What is the significance of buttocks?

From an interview with Shane Valentino, the Production Designer:

We ended up using John Currin's painting and I feel like in that scene specifically—she's all in white and has red hair and red walls—you can only use one piece in the background. There's a dialogue between her restraint and how explicit the painting is.

And, though you didn't mention it, an interview with Tom Ford about the meaning(s) and intention(s) of the nudity in the opening scene:

I don’t like a movie that’s about the art world that doesn’t have real art, because you can always tell. [...] I’ve lived in Europe for the last 27 years, so I thought, What do I want to say if I’m having a gallery show in America? Politics being what they are right now, I want to make a statement about America. And I remembered that great poster that I had hanging in my room when I was a kid, when I thought I was straight, of Farrah Fawcett in that red bathing suit. America was always tan, beautiful teeth, tits and ass. So, guess what? I want to talk about America today: Gluttonous, overfed, aging, sad, tired. [...] and [I] fell in love with these women, [...] I actually felt guilty that that had been my original intention. I found them so beautiful, so joyful, and so happy to be there. They were so uninhibited, and I realized that actually, they were a microcosm of what the whole film was saying. They had let go of what our culture had said they’re supposed to be, and because of that, they were so totally free. This is what’s restricting Susan. She’s being who she thinks she needs to be: I need to live this way, I need to look like that. [...] It certainly wasn’t fat-shaming — if anything, it’s a celebration of the beauty of their bodies, and a challenge to what we think of beauty, [...]

In terms of your question about why the focus on "buttocks", one possible answer is that for the nudity in a film directed by a gay man, the "buttocks" are the only sexually explicit part of the body that is common to both men and women. It could be seen as an non-gendered representation of sex and sexuality.

More abstractly, most of the film is about juxtaposition of the internal and external, just as the production designer and director said above. There's an intentional and persistent contrast against who the person sees themselves as, and who they outwardly project themselves as.

The central plot element of the novel inside the film is about Jake Gyllenhaal's character being unable to act in a manner to protect his family. Watching the slow and laborious playing-out of that scene shows that there was never really an insurmountable threat of violence, it's almost as if he just let the thugs drive away without doing anything. The point of the writing was that he saw the situation in that way of himself, but that he also rationalized that he was being sensible and reasonable in being calm and non-violent and non-confrontational.

Ultimately, like the artworks, like the nudity, like his own violent death in the end of the novel, and like the tumultuous end of his relationship with Amy Adams, there is a collective theme and question of how one is to act and/or present one's self rationally and reasonably in an irrational and unreasonably cruel reality.

The artworks throughout the film show a contrast of high artistry with common cruelty; the book included.

The focus on "buttocks" throughout the film is about exposure and softness and beauty, but how none of that really fits-in with the background/context.

When Amy Adams calls her daughter, it's at a moment in the film where this horrific scene has just started in the book, and it's a shock to find out that she actually has a daughter. Only later do we learn that she isn't Gyllenhaal's daughter, but it creates this stark juxtaposition of the lazy, Saturday morning nudity of young lovers in bed, carefree and presumably happy, against the terrifying violence and loss and chaos that Amy Adams is reading about (and that we are being shown) in the plot of the book.

Like the Production Designer said, the John Currin painting of the buttocks in her office creates contrast between her perfectly curated and arranged outward appearance, and the silly "vulgarity" of a renaissance painting that wildly exaggerates the "buttocks" via a convex mirror effect -- the work is called "Nude in a Convex Mirror" [2015].

When the wife and daughter are found in the junkyard, abandoned and dead on the red couch in stark contrast to their pale skin and red hair, their nudity again suggests the beauty of a renaissance portrait, despite the horrible violence they suffered and the cruel tragedy of their deaths. At first glance, the shot cuts in immediately and exudes beauty for an instant before the recognition sets-in that if they're naked and lying here in the middle of nowhere -- given all that has gone on -- then they're likely dead; which they were. It's a shock of beauty followed by the shock of horrible tragedy.

All in all, I don't think it's necessarily a specific recurrence of "buttocks", in the film, but a more specific recurrence of the juxtaposition of beauty, passion, love, and comfort against the harshness and cruelties and violence and reckless abandon.

In the end, we're left with an open-ended (http://www.vulture.com/2016/11/tom-ford-on-directing-nocturnal-animals.html) question about whether Gyllenhaal was too scared to show-up, got hurt or lost, or if he purposefully abandoned Amy Adams as a final act of revenge for how she hurt him.

Like the director's own statement about how his original intention for the opening scene was changed once he actually started putting his vision to film, I think the ending has a similar internal struggle played-out in the reality of the film. In a way, Gyllenhaal's character seems rational in his anger at the lack of support for his art, how she cheated on him, how she left him in the exact manner that her mother told her she would (despite her staunch refusal to be "that"), and for the abortion. His book portrays a sense of his self-envisioned weakness and inability to keep his family together and (quite literally, exaggeratedly) alive against the subtle, creeping violence of the harshness of reality. He ultimately gets his vengeance in the book, but also is killed by his vengeance. He can't get back the people he's lost, but by focusing on revenge for the loss, he also loses the only thing he has left: his own life.

In the end, like Tom Ford's self-revelation about wanting to satirize the gluttony of contemporary America but then being won-over by this uninhibited expression of shameless freedom, we can imagine too that Gyllenhaal's character creates the book and its narrative as a means of exorcising the "demons" of self-doubt and self-hatred and his anger towards Amy Adams' character's actions in their past.

He confirms the dinner date in email, but then he presumably never shows-up, or he shows-up the moment after the film ends. In a way, that ambiguity is supposed to be the instigation for an internal conversation within the audience members. Do you move on and never look back, or do you give things/people a second chance? What if they never changed? What if they became even crueler and colder and what if they're just letting you back-in, or reaching back out to you, because they have found themself alone and abandoned, even though they were the ones who caused that in you? There's even the argument to be made that their current loneliness doesn't compare the loneliness and pain they put you through in your past; so how much and how often do you expose yourself to them and give them another chance?

How do we, and how should we, navigate the juxtaposition of the cruelty and contrivances and chaos and caring aspirations of passions in reality?

Is nudity bad and vulgar, or is it beautiful and vulnerable? Or is it both?

What is life!?!