Rock stars are meant to seem smaller in real life. Particularly those who've had to shoulder as heavy a title as "saviours of rock'n'roll" for the past decade. But the Strokes – Julian, Nick, Fab, Albert, Nikolai – are enormous. As in, they take up an awful lot of space, particularly when crammed into a small lift. They're teasing their manager, Ryan Gentles, as he tries to take a call.

"Is that Barry?" Albert asks. He means Barack Obama. The others jump on board: "Has he heard the album yet? Ask him what he thinks of our album."

That album, a long five years in coming, is from one of very few contemporary bands still able to elicit wild hyperbole: it may be that even the president is excited about the Strokes' return. If not, he should be. Angles is their most adventurous record yet, forays into new territory – 80s synth, funk, even the odd shade of reggae – scarcely lessening the sense of exhilaration. Skittish guitar breakdowns and bitterly euphoric choruses will thrill the old faithful.

Their number includes the likes of Conor McNicholas, appointed editor of the NME a year after the band's debut in 2001. He calls the Strokes "one of the most important rock'n'roll bands of all time, without a doubt". Like many, he remembers exactly where he was when he first heard them: in his car, listening to the radio, on London's Stroud Green Road. "'Last Nite' came on," he says, "and it was one of those moments where you go: 'Fuck. Popular culture will never be the same again.'"

Quite a claim, but it's worth remembering how bleak the musical landscape was 10 years ago. Dance music was on a come-down and instead the charts were being battered by nu-metal. The Strokes, however, sounded like the best of the 70s reinvigorated. Jaded and scrappily exuberant all at the same time, they announced themselves as part of a New York continuum of rock that counts the Velvet Underground and Television before them. It helped, too, of course, that they dressed like James Dean, dripped with insouciance and somehow made falling out of Lower East Side bars drunk seem desperately romantic.

"We were just kids who had fairly decent taste in music," shrugs guitarist Nick Valensi, with a trademarked wide-eyed gaze. "And we were curious why no one made music like that any more, so that's what we did. It turned out that a lot of people had been wondering the same thing."

Among them was Geoff Travis of Rough Trade Records. He listened, so the legend goes, to just 15 seconds of "The Modern Age" down the phone before deciding to sign them. All true, says Travis.

"You hear that riff, the way the guitars start, and that's just classic rock'n'roll that you always want but don't hear enough," he tells me. "And the control – it was that lovely thing of perfectly controlled tension. It seems too easy but it's so hard to do. Rock'n'roll is not intellectual – it's primal, it's emotional. The Strokes just had it, really."

Though quintessentially New York, it was in the UK that the band really took off. They played Reading and Leeds festival in 2001 the weekend before their album was released. There was such demand to see them that organisers moved them from a small tent to the festival's main stage. When Is This It came out, the title seemed to function both as a sneer of ennui as well as a pre-emptive joke about anticlimax. Only, Is This It was all that: the album went platinum and remains a staple of "best albums of all time" lists.

"They reminded us what it could be all about," says McNicholas. "It was hip-swaying, tight jeans, boy-sex on stage and we hadn't had that in a long time." When he took up his editorship of the NME: "We had a new world and it had three pillars: one was our heritage in music, the other was the Strokes and the other was the [White] Stripes."

The White Stripes split earlier this month but it had always seemed as though the Strokes, "a dysfunctional family", in Travis's description, would be the first to go. After a five-year hiatus, beginning in 2006, in which every member but one (Nick) embarked on other projects, they appeared as good as broken up.

Today, though, there's no sign of tension, nor any indication that those years of excess have taken their toll. They still look great, if a little less shaggy. Guitarist Albert Hammond Jr just appears a bit vulnerable without his old mess of hair, his eyes more doleful than ever. Drummer Fab Moretti, who always seemed to have the sunniest attitude, positively sashays in to the studio in exquisite Phillip Lim trousers and some very LA shades.

Extravagantly side-fiving one another, they seem to hum with a forcefield of fraternal affection. There's also the literal humming of their mobiles: halfway through the photo shoot their single goes up on their website and the iPhone chirrups intensify. Albert looks up excitedly from a text at frontman Julian Casablancas: "Dude, our website crashed!"

He responds with a barely there nod and flicker of a smile. Julian – inscrutable, moody and charismatic – has always been the leader of this archetypal rock'n'roll gang and the band still tend to defer to him.

Albert for his part seems genuinely surprised. As does Nick, when I talk to him and bassist Nikolai Fraiture. "I still wonder why people make such a big deal about us," he says. "I listen to our records and I think: that's good, there's a couple of things that could be better. But to make the fuss that people do, especially in the UK… I'll be honest, I still don't get it. I dunno what it is… the gods of rock'n'roll are smiling upon us." Neither of them say anything for a moment. And then: "God, I really wish I hadn't said that."

Theirs certainly have been charmed lives. When Julian was 14 his father, who founded Elite Models, sent him to boarding school in Switzerland in an effort to curb his teenage drinking. It didn't kill his taste for booze, but at least he met Albert there. Nikolai was a friend from the Lycée Français de New York and then he met Nick and Fab, at the Dwight School, a smart Upper West Side establishment. By 1998 the five of them had come together and soon had a fervid New York following.

After Is This It came 2003's equally thrilling Room on Fire and by this point they could not really have been more famous. There followed the obligatory celebrity girlfriends: Fab and Drew Barrymore were together for five years and in 2009 Albert's romance with model Agyness Deyn was memorialised in a Valentine's Day spread for Vogue.

Their third album, First Impressions of Earth, came out in 2006, and the band had their first UK No 1. Perhaps inevitably, however, the band sounded diminished. Last year Julian told ABC News: "I was bummed that we weren't as good as I wanted [us] to be."

When they finished touring the album, Gentles confirmed that the band were taking a "much needed break". As Fab acknowledges: "Touring and drinking and doing drugs has its very definite toll, you know? Physically and mentally." When Albert thinks back to that time he says: "It seems like they want you to slowly kill yourself and write the best stuff ever possible, but there comes a point where one just takes away the other one. Living so fast you're not even doing music any more."

So it was a welcome surprise when, in 2009, Julian put out the stormingly good solo album, Phrazes for the Young. It's even more of a shock, though, that Angles finally got made. The first three albums were written almost exclusively by Julian, but this time they wrote together for the first time. "I think," says Nick, "it had to happen, otherwise we probably would have broken up."

It's thrillingly diverse as a result, but that way of working must have tested their diplomatic skills to the limit. "Some of us more than others," Nick says, uncharacteristically archly. Julian has been described as dictatorial, I venture. Does that have any truth to it? There's a long, heavy silence.

"Nnnno," says Nick slowly. "I don't think it's ever really worked like that. Over the years we all sort of fell into different roles and that happened very naturally. Julian never said: 'Hey, man, I write the songs.' I always felt pretty encouraged to bring in whatever I had."

I turn to Nikolai. Very quietly, he says: "I just don't know how to answer that question, actually."

Later, Fab puts it this way: "Julian was such a vocal and particular member of the band – the leader. I think he had a plan of stepping aside so that we would be forced to communicate more with one another." The band recorded the fourth album's songs without Julian, who provided vocals separately.

"What's rad about Julian," Fab says, "is that he knows what he's doing but he doesn't say it necessarily. I feel like we had so much shit to learn on this one, it was almost like the first record. Then, it didn't even occur to us that we had to follow certain guidelines. This time it's very apparent to us what we have to do and what barriers we have to cross."

Julian's reluctant to dwell on what those barriers have been. "It's hard talking about that past stuff because for me it's more about working towards what works," he says. "We could all probably say hurtful things if we had to figure out what went wrong."

I'm relieved Julian's talking at all: he's not had an easy relationship with the press and tends to be portrayed as a moody monosyllablist, sulking behind his sunglasses. First Impressions of Earth includes the track "Ask Me Anything", on which he sings, over and over, the words "I've got nothing to say": the lyric seems meant for journalists.

As the photos are being taken he hangs around awkwardly on the edge of the group, hands in pockets, like a teenager enduring a family photo at Christmas. He's facetious, yes (when I ask about regrets he sighs, "So many dead prostitutes"), but he's also as smart as he is silly and surprisingly eager to please. When later Gentles comes over to warn "one more question", Julian, like an excitable kid, bargains: "Three more! Two more?"

Perhaps the past few years have mellowed him. He, Nick and Nikolai are now all fathers and that, says Nikolai, "puts the important things in perspective – all the Spinal Tap bullshit goes out the window". There was, by all accounts, quite a lot of that. Theirs seems to conform to that crude rock'n'roll narrative: fame, followed by too much booze and drugs, followed by near-dissolution.

"And sex," says Julian cheerily. "Don't forget sex."

Other than that omission, he admits, "that kind of nailed it".

"It's funny," says Albert, "you don't want to live in cliche but it's just like, 'Nope, that was it'."

Nick calls some of it "really ridiculous". Now they're in their 30s he admits to feeling, "more motivated to make the band work and more grateful about being part of something special. I've exhibited a lack of grace many times in my life. I've had my moments, like, 'Wow, did I really complain that they brought me a Veuve Clicquot when I asked for Cristal?'"

The band's present-day relationship with booze and drugs is a bit more complicated. Nick says: "I'm a fan of drugs in general. I just think it's important to have some level of self-control. But, there are some types of people who just can't." Has he worried about his bandmates in that respect? "Yeah, but that's all I'm going to say."

Julian's more forthcoming. "I was living unsustainably and unproductively," he says simply. He stopped drinking around 2006, a year after he married Juliet Joslin, the band's former assistant manager. He's talked, in grim terms, about the three-year hangover that followed his giving up booze and hates the conflation of fast living with great music.

"I would prefer it if people thought that I didn't work hard, that I just played the guitar for three minutes a week and was like, 'Check out this song – what do you think?' That would be ideal. I would prefer telling people that I'm just truly talented. But you work so hard, you make it sound effortless. I mean, I never wrote when I was drunk. People just glorify that stuff so much."

"I was never much of a drinker," says Albert, "for me it was drugs." What he says next is a little cryptic, but he seems like a man struggling to express a new-found joy at being clean. "For me it's… I can't believe I didn't see it this way before. It's so hard to say the right words without it coming across cheesy or self-help. It's all so new. All the words are coming back to my brain. When you're fucked up you're just covering something else up you know? I feel like I've never played guitar better. I've never actually been better at, like… just being happy because of what I'm accomplishing."

Julian, looking on tenderly, nods: "Honestly happy."

"I feel lucky, actually, to have these guys and this thing," Albert continues. "I feel like it definitely… I'm just glad that I had it, friendship and music."

"Past tense…" notes Julian.

"Did I say that?"

"You said past tense."

Curiously, Albert isn't the only one to do this. They all seem to have a sense of Strokes-then and Strokes-now and this album as a turning point. As Fab puts it, with this newly collaborative approach, they've discovered a map by which to navigate. Whatever tensions they've had and have, their dynamic as a band "is still there, we jump right into it. It's like a specific way of talking, we have our own rhythm. I don't think that will ever go away."

When the others wave goodbye to Julian from across the studio he mimes some complicated goodbye gesture while making firearm noises. Are you shooting them with kiss bullets? "Yeah, it's a kiss machine gun," he explains, while reloading. His bandmates, recoiling in hammy slo-mo, catch invisible bullet-kisses to their hearts.

The Strokes' new album, Angles, is out on Rough Trade on 21 March

The Strokes were photographed at Skylight West Studio, New York