Viggo Mortensen is softly spoken, clean-shaven and casually dressed in jeans and a T-shirt. If you saw him in a restaurant, you'd smile at him not because you'd think there's a huge movie star, but because he radiates a gentle integrity and, well, niceness. But he's a disconcerting interviewee. The conversation goes like this. I ask question A, expecting answer B. He listens carefully, considers, and gives me answer E, and then we find ourselves on point K, V, or Z.

Luckily, we do keep returning to Everybody Has a Plan, a film that's close to his heart. Although it's the fourth he's done in Spanish, it's his first Argentinian movie. "It was like going home," says the star, who spent eight years of his childhood in the country. "The Spanish I speak is with an accent not unlike that of the brothers I play, and I knew the places from childhood." Everybody Has a Plan is a noirish thriller set in the lawless backwaters of Argentina's Tigre Delta: a tale of identical twins (both played by Mortensen) and stolen identity, with a tender love story at its heart.

Mortensen has long wanted to do an Argentinian movie. "I'd always thought I'd love to be able to say I'm part of its movie history," he says. And when the script – by first-time director Ana Piterbarg – came his way, he jumped at it. "It was one of the better ones I'd read in a long time: tight, well structured, very original." Such was his belief in it that he decided to produce it, too: "I wanted to help shepherd it, to make sure Ana's vision was taken into account." And, just for good measure, he even oversaw the English subtitles as well, ensuring they conveyed the subtleties that are so often lost in translation.

Playing twin brothers presented particular challenges, especially on such a low-budget movie. He tells me the scenes they appear in together were realised through old-fashioned tricks, namely using doubles and shooting on two layers. "But my other concern was to make sure they seemed like distinct individuals. We shot those scenes almost at the end – by that point, I had their body language, vocabulary, their way of being. It was actually fun putting them together."

The twins, as with so many of the roles Mortensen chooses, are opaque, strong, silent types. Broadly speaking, we have good twin/bad twin: good, urban-dwelling paediatrician twin Agustín has a breakdown and seeks escape in his brother's murky life. "In the end, he becomes almost another character," says Mortensen. "He accepts who he is: even though things are starting to go badly, he's at peace with himself."

He enjoyed playing the bad twin, Pedro, who lives literally and metaphorically outside conventional middle-class mores. "Pedro throws caution to the wind and he's incisive with his comments – he doesn't really care about coming off well, and he's dying of cancer, so he cares even less. Those sorts of people can seem crazy and frightening, or interesting and amusing – attractive even. You see people on the street yelling and think they're crazy, but maybe they're just happy and expressing what they feel at all times."

Mortensen, now 54, is not someone who likes to take the easy or obvious route into anything. A script has to be not only "good and different", but challenging. "If I think it's an interesting story, but I don't think I can do it – then that's probably a good sign.

"Each time I make a movie, it's like a paid scholarship to a different university course. It's up to you – you can just learn the script and certain mannerisms, props you might be given to play your part and do a technically great job, but I love the opportunity." He enjoys learning all kinds of peripheral things, he says, singling out 2011's A Dangerous Method. "Playing Sigmund Freud for David Cronenberg meant I read work by Jung, by Freud, remembrances, contemporary accounts. Right now I'm shooting a movie set in 1962 in Greece and Turkey." This is the Patricia Highsmith thriller The Two Faces of January, about a conman, his wife and a stranger trying to flee the police. "It's a very interesting period: what it was like then to be an American tourist in Europe is very different to what it is now." He's been studying ancient Greece, and tells me the crew were allowed to film in the Parthenon (normally, visitors can only walk its perimeter). "I was actually wandering around inside," he says, still marvelling.

The desire to be challenged has led to some striking roles this past decade. He was harrowing as the post-apocalyptic survivor in 2010's Cormac McCarthy adaptation The Road; he played a troubled academic in Nazi-era Germany in 2009's Good; and was a small-town family man who hides a dark past in 2005's A History of Violence. If his mission has been to shed the sword-wielding Aragorn image, he has succeeded; yet he's at peace with the fame the role brought him, acknowledging that he has The Lord of the Rings to thank for the opportunities that have since come his way. In Peter Jackson's Hobbit film, several of the Rings cast reprised their roles. Was he asked to take part? "No. Before they started shooting, back in 2008, one of the producers did ask if I would be interested. I said, 'You do know, don't you, that Aragorn isn't in The Hobbit? That there is a 60-year gap between the books?'"

Has he seen the new movie yet? No. "I'm interested in seeing that world again, and seeing what Peter Jackson's done, how he's made use of the improvements in special effects and cameras and the different way of shooting, which will probably enhance the visual aspect of the experience, the sound also. And I'm interested in seeing how he's managed to make three movies out of a relatively slim volume … Jackson is a clever person – I'm sure he'll have done something really interesting with it." Working in New Zealand from 1999-2003 remains an important professional and personal experience. "I've gone back occasionally to visit friends both animal and human. It's a beautiful country."

So if he retired tomorrow, what other films would he like to be remembered for? He considers. "The three movies I've done with Cronenberg are really good. Captain Alatriste in Spain – I'm really proud of that. The Road was a movie that has a good reputation, even though it wasn't released very well, but that's a movie I'm very proud of. And Ana's movie is something I really like."

Does he see a time when he might stop acting? He smiles as we chat. "In 2008, in this very room, I said to a journalist who asked if I had plans to do another movie, 'No, not at the moment. I need a break.' By which I meant a month or two. He then wrote, 'Viggo Mortensen has quit.' And I spent the next two years denying that I'd quit acting, or explaining why I had returned when I'd quit."

But for every exclusive Parthenon visit, there are press junkets to do, movies to promote and associated frustrations. "[The film industry] can be frustrating when people don't treat each other well, or when a movie isn't distributed or released well, or doesn't get the credit you think it deserves – which happens a lot," he says. He takes a dim view of actors who opt out of promotional duties. "Whether a movie turned out well or not, I said I'd do it, and I said I'd promote it. Fortunately, most of the films I've done over the last 10 or 12 years I've really liked to begin with, and I've really liked the way they've turned out, so – like Ana's movie – it's not a hard thing to talk about. Of course you do get tired though, and it can be very repetitive if you're doing it day after day, but it's part of the job."

He is characteristically vague about who he'd like to work with, although he does admit to a hankering for France – "the country that most respects movies and takes care of their own industry". I wonder if he has plans to return to his family's Danish roots and work in a country whose film and TV industry punches way above its weight. "Yes, sure, I'm interested. I've been offered quite a few things, including a Lars von Trier project. But the timings never worked out."

He has also turned his hand to script-writing, adapting a 1957 book by Mari Sandoz called The Horse Catcher: a coming-of-age story set in the 1800s about a Native American boy who must kill to be accepted as a Cheyenne man. "It's a universal story – it speaks about peer pressure to be violent. You could transpose it to an inner city and have instead the pressure to take drugs."

He hopes to direct it, too. "If I can get the money together. It will be a difficult film to make. You can prepare, but you have to be lucky, too." He reconsiders that remark and says: "There's a Persian saying, 'Luck smiles on the efficient.'"