MIA is having none of it. "I am not a conspiracy theorist," she says.

"Yes, you are," I say.

The great pop contrarian also known as Mathangi Maya Arulpragasam huffs. "What I said about the internet is what's happening now. It's on the front of your own newspaper. It's not a conspiracy theory, is it – unless your paper is supporting a conspiracy theory? Conspiracy theory is too much of a small pond for me to swim in." I feel suitably admonished.

It's been three years since MIA released her last studio album, Maya. Its first track was The Message, a 57-second discordant rap suggesting that social media companies were working hand-in-hand with the world's governments to spy on us ("Connected to the Google, connected to the government," she chanted repeatedly). A lot of people accused her of being politically naive. Now, following the Guardian's revelations about the spying capacity of America's NSA and Britain's GCHQ, it looks as if she was stating the obvious. Does she feel vindicated? "I do. I love it." She grins.

But after that third album, MIA decided she had had enough. She was bored with music, so made the gloriously titled mix-tape Vicki Leekx (a play on WikiLeaks) in two days, just to show she could, then quit. She moved to India with her baby son, Ikhyd, and wasn't heard of for ages. Now she's back, post-retirement, with a new album, so she can tell us about her retirement. Hard to get your head around? Welcome to Planet MIA.

We meet in east London, where she now lives. At 38, she still looks like a little girl: beautiful, garish, loud, a handful. She arrives with Ikhyd, now four, a sweet boy with huge brown eyes. The fact that he is here with her is a story in itself. For the past couple of years, MIA has been fighting a custody battle. Until recently, it looked as if she would lose her son to her former fiance, Benjamin Bronfman, unless she agreed to bring up Ikhyd in the US. As she says, tranquillity in her life tends to be transient at best, "the calm before the storm".

MIA prides herself on her normality. Her publicist tells me that she is one of the few stars who will turn up by herself, no fuss, just a regular civilian. But appearances can be deceptive. Sure, we're not far from the council flat where she grew up, but this is a fancy members' club with a pool on the roof and hipsters perched on sun loungers, cocktail in one hand, iPhone in the other. And she fits in perfectly: floral silk shorts, face-dwarfing shades, numerous bracelets, bags of confidence.

Within seconds, Ikhyd is crying. He has been told he can go swimming, but the pool has been divided into lanes for adults. She mops his tears and they strike a deal. Her mini entourage takes him off for a treat, while we're left to chat conspiracies, terrorism, even music, as if she's never been away.

MIA emerged on the music scene in the mid-2000s, the perfect antidote to confection pop. For a start, she wrote much of her own music, an unlikely, often inspired mash of rap, nursery rhymes, bhangra, electronic dance and punk. Music writers created a whole new lexicon to describe her sound, including the fabulous "gangsta shoegaze". But she railed against commercial success, and at the first sniff of a big hit – Paper Planes, which sampled the Clash's Straight To Hell, and made the US and UK top 20 – she recoiled. Yet she couldn't resist making headlines, whether for performing at the 2009 Grammys in a transparent dress when nine months pregnant, or giving the audience the finger at the 2012 Super Bowl. She is so much the perfect anti-pop star that sceptics have suggested her story is too good to be true: that in her own radical way MIA is as much of a brand as Madonna is. And, to an extent, this is true.

She says she grew up being told what she could do: as a girl, as the child of a single parent, as a musician, as a commodity. "If you're making music, don't talk about politics. If you're talking about politics, don't wear lipstick. If you're dancing in a club, don't talk about Sri Lanka." Sod that, she thought. "Actually, you can put the concept of freedom of speech next to rap music, next to my untouchable dad, who people talk of as a terrorist, next to random creative shit."

Which is exactly what she did. Her first album, Arular, named after her Tamil father, featured a song about refugees learning to say banana, which segued into the socialist dance track Pull Up The People, then into Bucky Done Gun, a boastful rap announcing her arrival. You could see her art school background in the attention to detail, in the way she dressed: the pastel green eyeshadow, fuchsia lipstick, neon prints. At times, she looked like a Fauvist painting. She has collaborated on a string of brilliantly cinematic videos: Romain Gavras (the son of film-maker Costa-Gavras) directed Bad Girls for her, with its high-speed car chases, and before that Born Free, which tackled the issue of ethnic cleansing by showing redheads being rounded up by police and taken to a desert to be shot. It was meant to stoke controversy, and sure enough YouTube refused to host it. She has had surprisingly few hits (Paper Planes is her only really successful single) and many have failed to chart, yet Jay Z, Kanye West and Madonna, all far more commercially successful artists, have wanted to collaborate and borrow some of her credibility.

We're sitting on a hanging sofa in the shadows, and MIA is swinging away happily, drinking coffee. It's the last thing she needs: she is talking 15 to the dozen about everything she's been doing, from stressful court appearances to painting and finding herself in India. She talks excitedly about how she became interested in Hinduism and explored the origins of her name, until everything began to make a weird kind of sense. Her music has always been about family: a search for identity that often raises more questions than it provides answers. Her second album was named after her mother, Kala. Her third album was Maya, the name she grew up with in England and still goes by (it's easier for English people, she says). And now Matangi, her birth name.

On stage in Brixton in 2010; with former fiance Benjamin Bronfman; pregnant at the Grammys in 2009; in the Bad Girls video; and at the 2012 Super Bowl. Photographs: Getty Images; Rex Features; Planet Photos

She was born in Hounslow, south-west London, in 1975. When she was six months old, the family moved to Jaffna, a Tamil town in Sri Lanka. She barely remembers her father Arul, even though she has read plenty about him. He left to join the struggle for Tamil independence, and she and her two siblings were brought up by her mother. They went into hiding from the Sri Lankan army, then moved to Tamil Nadu in India. Just before her 11th birthday, they came to England as refugees; her mother worked as a seamstress for the royal family.

She says that whenever she asked her mother why she was called Mathangi, she got a vague answer about being named after a newsreader. She didn't think any more about it until she became interested in Hinduism. "When I typed in Mathangi, I found out Matangi [without the h] was this goddess of music, and the second link was me, MIA, rapper. I thought, this is funny, and within an hour there were so many things that linked up all these things." She talks about these coincidences or not-quite-coincidences as if they were divine revelations. "Matangi's mantra is aim, which is MIA backwards. She fights for freedom of speech and stands for truth, and lives in the ghetto because her dad was the first person in Hindu mythology who came from the 'hood, but had gained enlightenment through not being a Brahmin."

Did she shout, "Mum, I've worked out why you called me Mathangi: I'm the goddess of music!"?

She looks at me as if I've totally missed the point. "No, my mum and dad are Christians."

So why has Hinduism been such an influence? "It isn't. I Googled it." She goes off on one about Brahmins, untouchables and the caste system: all the reasons she could never embrace Hinduism.

What she has embraced instead is a homemade belief system, a potage of pyramids, squares and circles. "I devised this theory that the world is made out of three types of people. Squares are logical and scientific. They create tools, so Steve Jobs would have been a square. The pyramids are the money structure and the religious structure and the ideological structures built by human beings. It's always a pyramid: the guy at the top, billions of people on the bottom, and you have to claw your way to the top and kill a lot of people on the way. The circle is all the other stuff we can't explain. If you cut a tree, the rings are circles, your eyeballs are circles, cells are circles. Einstein discovered how to split a circle and it resulted in the most negative thing for humankind. You can't fuck with the circle. And to be a successful human being, you have to understand all three concepts."

Blimey. So she's created a new religion of squares, circles and pyramids? She swings away enthusiastically, and looks pleased with herself. "I did. I justified the world." Where does she see herself? She gives me a withering look, astonished at my stupidity. "No, I said you have to have knowledge of all three. I use my computers to make my work. I'm in the music industry, which is a pyramid, and I'm a human being. If you cut me, I've got round circle cells."

What she really wants to know is, who and what is at the top of the pyramid. "I don't have an answer. You can only ask the questions to the guy at the top." Why not go around the world, asking all the people at the top? "I'd probably get killed in the first 24 hours. They'd be like, 'Oh, yeah we'll show you!'" She's joking, and she's not.

Actually, she does have an idea about what's at the top. Space. "What is the promise? That we go to space? How many people will go to space and colonise another world? The person who gets to the top usually has to be a cunt to get to the top. So they are going to get on a spaceship and go to another planet and populate a world, so why would you want to live there?" My head is spinning. She's philosophised me into a state of confusion. She grins. "Confusion is good, isn't it?" In a way, she says, that's what all her work is about. "It's a landscape of confusion." Do friends find her pronouncements bewildering? Do they go… She's already laughing and answering the question: "'Shut up.' Yeah, that's the thing. Half the time people think I'm high, but I'm not."

If MIA thinks the powers that be are out to get her, it's not entirely surprising. When she criticised the Sri Lankan government in January 2009 for its alleged use of chemical weapons on Tamil civilians, the foreign secretary, Dr Palitha Kohona, said she should "stay with what she's good at, which is music, not politics".

If she is combative, it's because she feels she's spent her whole life fighting. She was turned down when she applied to study art at Central Saint Martins, but when she told them the decision would ruin her life and she'd end up a "crackhead prostitute", they let her in for sheer chutzpah. She started out making videos for her friend Justine Frischmann's band, Elastica. It wasn't until she went to Sri Lanka to make a film about her cousin Jana, who had gone missing in action, that she morphed into MIA. "The day I graduated, I heard about Jana. There was a story that he'd died. Then somebody said, no, he's alive, but he's a vegetable. Then somebody else said, no, he's just MIA. When someone in Sri Lanka joins the movement, you never know what happens to them. I grew up with him for 10 years; he was like my twin. So the first time I used MIA was when I was looking for him." The film turned into an exhibition, and then into her first album, Arular.

When she made Arular, she contacted her father and told him she was going to name it after him. He forbade her. Why? "I don't know." So what did she say? "I was like, it's the only thing you gave me: I'm going to fucking use it."

She sounds awestruck when she talks about her father. She says he was sent from Sri Lanka to the Soviet Union to study as a teenager, where he thrived in a harsh environment. "My dad grew up in a mud hut and studied by candlelight. He was 14 when he got a scholarship to Russia. He was super clever – the cleverest person. He landed in 5ft of snow, and was alone at 14, studying science and engineering. He didn't have a bed and he slept on a table." It sounds like a novel. "He was tough because he was poor, and I can't imagine Russia being tolerant to brown people in the 60s or 70s. It can't have been a walk in the park."

Yet at times she seems to despise him, too. In the trailer to a fraught, much-delayed documentary about MIA, leaked this summer by its director, Steve Loveridge, we see footage of a younger, lisping Maya talking to a camcorder. "This is what happens to a kid whose dad went off and became a terrorist. And this is how it fucked up the family… He was one of the five original members who founded the most lethal terrorist organisation the Tamil Tigers." However much she condemns the hypocrisy of traditional marketing, she knows her backstory has cachet. She has denounced others for calling him a terrorist, saying he was a freedom fighter in Sri Lanka's non-violent revolutionary student movement Eros. Which is true? "He was a member of Eros. It was an intellectual movement." So why does she call him a terrorist in the film? She didn't know any better, she says: "How many 20-year-olds do you know who know of individual struggles, and the names of each group? The media brand every struggle against the state as terrorism." The reality is that she's not sure; there is often no simple demarcation between peaceful and violent protest within a revolutionary group.

'I don’t consider myself a musician. I’m an artist.' Photograph: Jason Evans for the Guardian

Anyway, she says, the bottom line is he did screw up the family. In Sri Lanka, she was told that she wouldn't make anything of herself because she was raised by a single parent. There was more of this when she came to Britain. "At school, teachers said you're going to be fucked up, you're going to stack shelves at Tesco because you don't have a dad. And statistics say that 80% of kids from single-parent families are losers. So by the time we got to where we got to, we realised we had to fight 10 times harder." In the event, she and her siblings all did well for themselves; her sister, Kali Arulpragasam, is a successful jewellery designer.

Is she angry with her father, or proud of him? "Neither. Just curious." Where does she stand on Tamil independence? "I support Tamil people, of course. I always have. Whoever sticks up for Tamil people will have my support." Is she still in touch with her father? "No. No. I guess I'm like him a bit. A bit of a loner. I like to do my own shit, then set off." Look, she says, it's no big deal. "We don't come from a culture where we have to have group therapy. It's just a very dysfunctional modern family. That's what we are." As for her mother, they couldn't be closer. "She's taught me how to be creative, open-minded, loose and strong. Strength in a different way. Strength through endurance."

Given her relationship with her father, the thought of losing Ikhyd must have been terrifying. I ask about her custody battle with Bronfman, whose father ran Warner Bros. She says she can't really talk about it, but of course she does – and with some anger. "In America, it's not the mother who has the say. The legal system sides with money. It doesn't side with truth, it sides with who has the best presentation of truth. I guess it was a lesson I had to learn. So you just had to be really good at lying. And I'm not very good at lying. I just can't be bothered. When it comes to basic shit, like a mother and child, you have a huge judicial system that no longer backs them up."

If that's the case, why couldn't she have simply bought the best lawyer and won the battle with the best arguments? "I'm not going to go against the billionaire with my money. Ultimately, I'm a working mum. I don't have parents who have tons of money who can fund my shit for 20 years." Is that what it came down to, going up against a billionaire? "Well, it was the idea that I was wild, a rock star mum, versus a financial institution, and fighting it in a country where financial institutions ultimately have the power. As a mum, my experience of the law in America was disgusting, because mums don't have a say. And I'm not going to change that, because it's going to take a nation of mums to stand up for it."

She says the irony is that after 9/11, she found it difficult to get into America, because of her support for the Tamils. Then, after she and Bronfman split up, she and Ikhyd found it impossible to leave; there was thought to be a risk she would abduct him. A few months ago, she finally won her battle and her relief is still palpable. In June she tweeted: "FREEEDOM > ME AND IKHYD IN LONDON TOWN! HOOOOOOOOOO MMMMMEEEEEEEEEEEEEEE!!! NICE TO BE OUT OF DARKNESS."

Has her experience put her off men? "No. I'm in a relationship and have been for some time, and I'm happy." In fact, she's more than happy: "I'm in a state of nirvana." Nothing is going to spoil her mood. Not arguments with her record company, who have delayed the release of the new album to the point where she has threatened to leak it herself. Nor fallouts between Loveridge, the director of her documentary, and her management ("Count me out," he told them. "Would rather die than work on this"). Now the NFL is suing her for $1.5m over her middle-finger at the Super Bowl, but she doesn't care about that, either. "It's a massive waste of time, a massive waste of money," she says in a video posted last month on YouTube. "It's ridiculous… a massive display of powerful corporation dick shaking." It's also an example of the hypocrisy of the music business, she says, pointing out that while she is scapegoated for her "display of female empowerment through punk rock", the NFL was happy to have cheerleaders "under 16… hips thrusting in the air, legs wide open… in a very sexually provocative position".

Misogyny and the exploitation of women are a recurrent theme in her music. What does she think about Miley Cyrus and her twerking? "It's a spectacle," is all she'll say. "I'm not talking about anyone else." She has more respect for Madonna's brand of self-exploitation, and tells me a funny story about a song she offered her from the new album, Sexodus. "I actually played that song to Madonna when I wrote it and said, 'Oh, you can have this.' She gave it back to me."

Madonna didn't think it was right for her, but MIA still wanted to know what she thought of the question behind it: when you have it all, what should you do with it? "So I was like, what d'you do? And she said, 'You spend it!' And I said, no, I don't think that's the right answer." Did it change her opinion of Madonna? "No, because Madonna is true to Madonna. She always said she was a material girl."

Is the delayed release of the new album down to conflict with her record company, Interscope, or the result of personal problems? Time for another conspiracy: "Both. I thought, it's funny the shit in my life started only when I handed my album in." The custody battle? "Yes, my personal life, and it's really hard for me to separate them."

My own experience suggests MIA is probably as responsible as the record company for the delays. In August, I was sent an early copy of the album, but after a couple of days had it whipped away from me, "for minor tweaking". I didn't get it back until two months later, and it was noticeably different. There's a bit of everything, from Hindi punk rap to the self-referential Boom Skit about her time in America ("Brown girl, turn your shit down… Let you into Super Bowl, you try to steal Madonna's crown"), to the gorgeous lushness of Sexodus, a collaboration with The Weeknd. Then it was whipped away again.

All she wants to do is get it right, MIA says. After all, so many things have turned out fine: she's back in London with Ikhyd, she has a new love, a new record and a better understanding of her place in the music industry. "Nirvana has nothing to do with making money, nothing to do with having a number one single on Billboard, nothing to do with me sucking anyone's dick." Was there a time when she thought she'd have to go down that route? "As a human being, you're bombarded with that image every day." Is it hard to resist? "Yes, of course it is. The only way I could do it was because I came from a fucked-up situation, so I knew how to survive on a fiver – I tasted real struggle. But if you're a kid that's grown up in England in a semi-OK environment, it's difficult. You're going to have the fear, aren't you? The assumption of darkness is scary, and that's what it works on."

Meanwhile, MIA is content to plough her own furrow. She fancies going on The X Factor in disguise. "I was thinking of wearing a burqa and auditioning." How far would she get? "First round? No, second round. They'd give me a second go, because I'd be in a burqa and they'd be politically correct. Then I would get kicked off." Does she think she's a good singer? "Course not. No! I'm just a… not really a singer, I don't know what I am." Is she a musician? "No, I don't consider myself a musician. I'm an artist." In her documentary, she has a pretty good go at defining herself: "I could be a genius, I could be a cheat," she says. "It's a thin line and I'm fuckin' with it."

By the time you read this, MIA may well have retired again. "I don't know what comes next," she says. "I guess I'll have to live a little before I decide." How would she feel if Matangi went to number one? "Matangi lives on the outskirts of the mainstream. It will never go in the charts." Would she be appalled if it did? "Pretty much, yeah."

• Matangi is released on 4 November, on Neet/Virgin EMI.