Gyllenhaal stresses to me the universality of Brokeback's story ("My character could have been played by a woman and it would have made just as much sense," he says), but I'm astonished when he says that he doesn't believe Ennis and Jack are gay. "I approached the story believing that these are actually straight guys who fall in love," he says. "That's how I related to the material. These are two straight guys who develop this love, this bond. Love binds you, and you see these guys pulling and pulling and tugging and trying to figure out what they want, and what they will allow themselves to have."

One of the film's producers, James Schamus, is as surprised as I am when I tell him that Jake perceives his character as straight: "Did he really say that? Well, I suppose movies can be Rorschach tests for all of us, but damn if these characters aren't gay to me. I think what Jake might have meant is that these guys lived outside of a social construction of a gay identity. There was no such thing as a gay identity for a cowboy in 1963."

If you believe the rumors in the blogosphere, Gyllenhaal might be looking for his own gay identity. In the month before I met him, two seemingly conflicting rumors circulated. The first claimed that Gyllenhaal gave way to a body double for Brokeback Mountain's nude scenes. The second said that he is bisexual and looking for an opportunity to come out.

Gyllenhaal flatly denies using a body double. As for his sexual orientation, he says this: "You know, it's flattering when there's a rumor that says I'm bisexual. It means I can play more kinds of roles. I'm open to whatever people want to call me. I've never really been attracted to men sexually, but I don't think I would be afraid of it if it happened."

The day after our dinner, Gyllenhaal invites me on a walk with him and his German shepherd, Atticus, in Runyon Canyon, a 160-acre park near the Hollywood Hills. It's a sweltering morning, and soon enough Atticus and I are panting and looking for shade. Eventually we come to rest on a bench facing a huge hill in the distance. Atticus scoots under the bench.

I have yet to ask Gyllenhaal about Kirsten Dunst, and I figure that this is as good a time as any. But he'll have none of it. "I don't want to talk about that," he says politely. When I ask him why, since he used to talk openly about their relationship, he says that was "before there was such an insane interest in it."

Indeed, Jake and "Kiki" (Dunst's nickname) inspire only slightly less rabid interest in the gossip rags than Jessica and Nick do, and the fever only spiked when Gyllenhaal and Dunst, supposedly no longer a couple, recently began appearing everywhere together, including attached at the lips poolside in L.A. (If one celebrity magazine's "body-language expert" is to be believed, the couple is not only reunited but "very much in love.")