Agyness Deyn: 'Who am I?'

I don't know if this is a testament to the alchemy of fashion photography, or to the visual versatility of this particular supermodel, but my first thought when she walks in is, oh, here's the makeup artist. Or the personal assistant, perhaps. Whoever she is, she's bang on time – so in the tradition of supermodel folklore, maybe she's come to announce that her boss will be a gazillion hours late? Then the woman offers a smiley greeting in a soft Lancashire accent, and I realise she must be Agyness Deyn.

She doesn't look anything like Deyn. She doesn't even really look like a model. Which is weird, because Deyn was the first British model since Kate Moss to break through from the catwalk into national consciousness, her trademark tousled blonde crop so recognisable that for a while young women in salons across the country were demanding an "Agy". Named model of the year at the British fashion awards in 2007, and honoured by an entire issue of i-D devoted to herself, she has been the flagship face of Burberry, Vivienne Westwood and Giorgio Armani among many others, while subverting the luxury brand aesthetic with a look best described as army surplus meets jumble sale chic. Deyn has dated rock stars, sung on a Rihanna music video, and defined a version of cool that style magazines invariably like to describe as edgy and fierce.

The woman I meet isn't fierce at all. But then, she's no longer a model either. She is incredibly polite and attentive, to the point where she can come across as almost meek, and talks slowly and carefully, the words punctuated by long pauses while she considers her thoughts. After modelling for 11 years, Deyn has decided to swap the catwalk for the stage, and this week will appear in her first play, a four-handed comedy called The Leisure Society. It is, she readily admits, quite a change.

"For modelling, you have to be such a strong person in a way – or seen to be a strong person, do you know what I mean? But in acting you have to get in touch with all the vulnerability that you carry." She's been rehearsing for eight hours every day, and talks about it with a sort of awestruck reverence.

"When I started doing little bits of acting I was like, this is what I'm supposed to be doing, you know? Going to this place where there's this fulfilment in kind of like letting go, and also that part of really getting in touch with yourself, to be able to understand a different character, and then portray that to whoever's watching it, and to be able to penetrate them in a way that touches them. And there's this unsaid kind of bond with us all [in the cast] which is so powerful, you know? It's like this family, there for each other; I feel held by the other people who are doing it, which is so great, cos like with modelling for so long it's a very solo project, and quite lonely. Being part of – well, it makes me feel emotional just talking about it – being part of this with these great people – it's just like so cool."

Deyn plays Paula, the 21-year-old date of a thirtysomething divorcee who takes her to dinner to meet a well-heeled couple he has been friends with for years. The couple had planned on ending their friendship with him that night, but Paula's presence proves to be sexually explosive; she and her date describe themselves as "special" friends, an arrangement known to fans of Sex and the City as "fuck buddies", and the evening takes an increasingly bizarre series of turns as the older couple's attempts at worldly sophistication unravel into erotic confusion. I've read the script, and it's extremely funny.

"The writing is genius," agrees Deyn. "It's a back-and-forth of absurdity. Paula's 21, and I suppose when you're 21 you have no filter, you have no concept of, like, consequences, but it's not like a conscious thought – everything is a bit black and white, cos you've just not had enough experience. When you're young you say it how it is, and even your views are, like, 'This is totally the truth', cos you don't know any difference, so there's a real confidence in your way of thinking."

Was she like that at 21? "Yeah! And I kind of feel like I have to put me as I am now to one side, cos you know, I'm 29, I feel like that's the age when you start to think about life. What is this all about? Who am I?"

Even simply being photographed as herself, no longer as a model being paid to sell something, can be confronting. "When you're doing it as yourself it's a bit different. When you're doing it yourself, it's, OK, this is me. And I suppose it does make you explore it; am I OK being who I am?" Does it pose the question, who am I?

"Yeah, definitely," she agrees, her eyes widening. "So it's really different. But really nice too. I suppose it's a challenge, but also a relief."

Deyn seems to become less comfortable when she talks about her old career as a model. She's the first professional beauty I've met who doesn't pretend to have been an ugly duckling – "No, I suppose as a little girl I was really pretty. I was cute" – which makes a lovely change. But apart from that, she appears to be bound by the fashion world's omertà, which prohibits any acknowledgement of the industry's notorious darker sides – bitchiness, eating disorders, insecurity – or of any ambition to be a model at all, a career that convention dictates must arrive by surprise, like a baby delivered by a stork.

Deyn was born near Rochdale in Greater Manchester, to a postal worker and a nurse, and though they divorced when she was 12 her childhood sounds happy; a normal provincial upbringing, with a part-time job in the local fish and chip shop. Her great childhood friend remains her best friend today – Henry Holland, now a successful fashion designer – and in the past she's said that as dreamy teenagers, "Henry would say: 'When you're a famous model', and I'd say: 'When you're a famous fashion editor.'" But when I ask her now why she'd wanted to be a model, she says: "I'd never really thought about it until I got spotted." Hadn't she won a local modelling contest at 16? "Yeah but it never crossed my mind that it could actually be a career." What else had she been considering? "Who knows, who knows," she offers vaguely. "Don't know."

At 18 she was spotted by a scout in London, while visiting Holland at university. For the next five years she worked as a jobbing model, not making much of a name for herself but having a ball; another childhood friend joined the pair in London, and they were like, she grins, "the three musketeers". It wasn't until a New York agent signed her in 2006 that she suddenly became an overnight star, filling the vacancy of Britain's new supermodel, previously occupied by Kate Moss.

"That still like blags my head when people say that. Even now I'm only just thinking like that. It was really mental." There must have been a moment when she knew she'd joined the ranks of supermodels? "No, I never really clocked on to that." Even when her hairstyle had its own name? She looks fleetingly embarrassed. "I suppose, yeah, there was a bit of that. I think in the moment I had realised," she reluctantly concedes, "but I don't know whether not really accepting it was a way of actually not getting caught up in it. Maybe."

When I ask how much truth there is in the bitchy reputation of modelling, she replies cautiously: "Um, my experience is that I just went to work and I had fun." A long pause. "And then I went home, and then I wasn't at work any more." But there was quite a bit of bitching about Deyn, because gossips claimed that when she became fashion's sensational new star, she knocked several years off her age to disguise the fact that she had already been around for ages. When I bring this up she freezes, her face a blank guard, so I ask if it's true, as I'd read, that she resorted to bringing her passport to interviews to prove her age. She relaxes and laughs.

"Really? That would have been clever. No, when I decided I would really do modelling I was like 18, and I think at the time that was quite old for a new face, so we knocked off a few years." The deception has long since been corrected. "But it was my birthday last week, and Henry was saying how old would you have been? Cos it got really complicated – when personal and work collide – and Henry was like: 'But it looks like I started being friends with you when you were four or something!'"

She seems to tense again when I ask about the other curiosity in her biography. Deyn was born Laura Hollins, and according to which account you read, she changed it a) because her agency told her too many models were already called Laura, or b) she and her mother saw a magazine cover that said "Change your name, change your life!", so they both did, or c) she consulted a man who claims to be Britain's only "name analyst", Laurence Y Payg, and uses a 3,000-year-old Chinese technique to create a more "positive" name. His technique sounds a lot like numerology – spelt with its superfluous Ys, her new name apparently equates to the number 21 – and Payg has publicly credited it with all Deyn's subsequent success. As I run through these explanations, she gazes back with an unreadable expression that could be hiding annoyance, confusion, amusement, or almost anything. Can she explain what happened?

"Yeah, in two seconds. Yeah, I started modelling, and they said there are loads of Lauras – you can change your name. I was like, OK, and then I thought about it and said OK, I'll be called Agnes; my grandmother was called Agnes." It was as simple as that? "Yeah." But didn't her mother, a Reiki master, also change her name – and Deyn's sister too? Yes, she acknowledges warily. And she changed the spelling because of numerology? "Laurence helped me change my name," she says matter of factly. "Erm ... yeah, that's it."

I think she means that a), b) and c) all happened, in that order, but when I point out that numerology is quite an esoteric notion, she looks blank again. "I suppose I'm open to trying new things, do you know what I mean?" I quote Payg to her – "Soon after her name change, Agyness's career sky-rocketed. Absolutely nothing else about her circumstances changed ... and yet everything suddenly changed." Does she really believe that's why it happened? "I don't know," she murmurs, looking uncomfortable. "Who knows?"

I get the impression that Deyn finds interest in her private life quite difficult to deal with, and would rather come across as vacant than revisit subjects which have left her feeling bruised by the media. She put her $2.5m New York apartment on the market recently, and photos of its wildly baroque, lavishly idiosyncratic interior (conch-shell chairs, a coral and seaweed mural, leopard skin carpets, lion's head taps) caused quite a sensation in the press. When I ask how that felt, though, she claims to be unaware of any coverage. "I did the decor with a friend of mine, and had a lot of fun doing it. But it's for sale now, and that's it really."

I assume she must be fabulously rich, but when I say so she looks astonished – which I think is another way of seeing off unwelcome intrusion. "Well, I suppose I've never really had a lifestyle that needs upkeep. I don't get cabs; I'm on the tube with my Oyster card. I've never really changed my lifestyle."

But at moments she can also be disarmingly vulnerable. Having dated Josh Hubbard, guitarist with the British band the Paddingtons, and then Albert Hammond Junior of the Strokes, she is currently single, and volunteers: "Yeah, single for three years now. I think I've forgotten how to function in a relationship. I hope I meet someone, cos it would be nice to have someone to share your joys and accomplishments and hard times with. To have a comrade in life. I'd like a relationship that was like two tree trunks side by side, strong but independent."

It's quite hard to reconcile her fiercely edgy supermodel image with this sweetly shy, almost earnest ingenue, and she agrees: "I think I was able to put on more of a brash front a lot more when I was younger. Now it's a bit harder, so now I kind of embrace the feeling awkward, maybe a little bit socially inept. But in a good way." But just as I'm leaving, I get a glimpse of the person I would guess her close friends know. She explains that she is currently living here in the five-star Corinthia hotel where we're meeting – and meets my look of surprise with a playful grin.

"Yeah, I love this place. I was sat outside chilling out yesterday – I feel so chilled here, I actually feel like it's my house – like, what are all these people doing in my house?" she jokes. "And I was talking to this chap, chilling out after rehearsing all day, and we were just chatting and stuff and I didn't know he's the owner, the guy that built it – and he lives here – and I'm like going," and she starts to laugh, "S'all right this hotel, isn't it?"

The Leisure Society is at Trafalagar Studios 2 from 28 February to 31 March. Tickets are available from atgtickets.com/trafalgarstudios