Tilda Swinton didn’t think she’d had a prayer of winning the Oscar when she attended the ceremony in 2008 for “Michael Clayton.” After her name was announced, the shocked actress stumbled to the podium and promised to give the statue to her agent Brian Swardstrom. But first, she took her Oscar back home to Scotland, because she wanted to show it to her gardener (he’d never seen one), before handing it off a few months later. “It’s on my dresser in my apartment,” says Swardstrom, who tried not to accept the gift. “I have willed it back to her children, so it’ll go to the Swinton household at some point.”

Swinton is back on the awards trail this year, with Wes Anderson’s “Grand Budapest Hotel” and Jim Jarmusch’s “Only Lovers Left Alive” (which landed her an Independent Spirit Award nomination last week). Her most memorable turn, and the one generating Oscar buzz, is as a dystopian tyrant in Joon ho-Bong’s “Snowpiercer,” the Radius-TWC release that scored a coup for on-demand distribution by grossing an estimated $8 million so far on VOD.

Swinton is happy that audiences have discovered the action film. But the actress, who is in New York to accept a career tribute at the Gotham Independent Film Awards on Monday night, has been urging Radius to give “Snowpiercer” another try in theaters. “All the readers of Variety should lobby for them to re-release it, so people can see it on a big screen,” Swinton said over a dinner of lobster sausage and a virgin mojito at MoMA. She talked about the indie film business, Oscar campaigning and why she doesn’t own a TV.

When I first saw “Snowpiercer” and Mason entered, I laughed, and then wondered if it was you.
I’m delighted not to be recognized.

How did you hear about the role?
My friend Joon-ho Bong pitched it to me by saying, “We know we want to work together, but there’s nothing for you in this one.” The part I play is still written in the script as a mild-mannered man in a suit. Then he said, “What about trying to fiddle with that?”

Did the part get bigger?
No, but we never changed his gender. You may noticed that Mason is still referred to as “Sir.”

How do you build the look?
I thought of all those bombastic leaders. They just seem to get more and more outrageous, like Mussolini or Gaddafi or Hitler or Margaret Thatcher. They overact like nobody’s business. And they are completely over the top. We’re constantly saying, “Isn’t George W. Bush ridiculous?” There’s a tendency of trying to picture these monstrous despots as humorous. Bong and I thought how it needed to be grotesque. We just dressed up like it was Halloween.

Did the wardrobe come first?
We found a photograph of a woman—I think she’s a woman, I’m actually not sure. It was a person who had glasses and a certain kind of stoop and a similar hairstyle. That was our starting point, and I drew on certain other references that amused me. I got a piece of silly tape and stuck my nose up. We wanted it to be rough. The wig was never glued or pinned on. It was like a hat. The teeth are super fake. I think they come out at one point. I’m not even sure what gender Mason is.

How do you feel about Oscar campaigning?
I still don’t know what it is. I did once go to that ceremony with a film [“Michael Clayton”] that was on that track, but even then, I didn’t really know what was going on. That’s my privilege. I live somewhere else. I’m here now because this film and that filmmaker is really important to me. It still hasn’t been seen by enough people, so any opportunity to get its name mentioned is significant for me.

Did you think you were going to win?
No. Did anybody?

You’ve been in both studio movies and indies. How has the business changed?
Truthfully, I’ve only made a handful of studio movies. I had a part in “Benjamin Button,” the David Fincher film, but it was a small part. My experience in the last few years has been entirely in independent films. I have worked with filmmakers, for example, Jim Jarmusch, who has been making independent films for a long time. If he was sitting here, he’ll tell you it’s tougher now for him. I hate to think of that, particularly since with “Only Lovers Left Alive,” you’d want it be easier for him.

What makes it so hard?
My analysis is that it’s the crisis in distribution. Having said that, people are trying to reinvent the wheel in intelligent ways. There’s Tom [Quinn] from Radius sitting right there. If the choice is not seeing a film at all, or seeing it in a new weird way, all the power to the inventors. I would say distribution is the area where people have to start really thinking creatively.

Do you have VOD in Scotland?
I don’t know. I don’t have a television.

How do you watch movies?
I go to the pictures as often as I can. And we have a projector.

What’s the last movie you saw in a theater?
Kubrick’s “Parts of Glory” in a theater last week. The one before that? The “Lego Movie.” I loved it. It’s a brilliant film.

What do you make of the lack of good roles for women in Hollywood?
I have two ways of coming at that. I recognize that in the audience, absolutely. As a film fan and a woman, I am constantly looking. We look for our avatars in the cinema. It’s nice for women—and for men—to put themselves in the shoes of a female. But as a practitioner, I have to confess it’s not an issue for me. I tend to make my work in development with filmmakers.

You work with your friends.
I don’t necessarily feel cast by them. I feel we tend to dream these things up together. In terms of looking for opportunities, I’m very lazy. I’m all for homegrown. I don’t go to the market.

When was the last time you had to audition?
I did read some scenes for the Coen brothers for “Burn After Reading.” Because I wanted to. That’s a script so precisely written, you sort of need to hear it in your ears and feel it in your own mouth and check that it’s making the right sound for them.

Do you ever Google yourself?

Why not?
Because I have too much to do.

What about Twitter?
Life’s too short. And I’m a Rip Van Winkle in that way, I’m afraid.

You know that people consider you a fashion icon.
I’ve heard. It’s not something I set my cap at.

Had you always been fashionable?
I don’t think I am! I’m trying to think how old this jacket is. Seriously, let me think. I got it 10 years ago, because I had this when I met my sweetheart. That’s not very fashionable.

You just got in from Paris.
I was making a performance piece for the last two weeks with a friend of mine. We’ve made three of them now. The first one we did was called “The Impossible Wardrobe,” which was a fashion show in which I took pieces from the museum’s collection—like Napoleon’s jacket—and showed them to the audience in a mirror. The second piece we did last year called “Eternity Dress,” where we made a dress during the performance. This was one was called “Cloak Room,” in which I invited the audience to give me their coats.

Did you keep them?
No, we gave them back.