The relationship between Squall and Seifer is one of the most interesting character dynamics in Final Fantasy VIII. There's a lot going on there! Even when Seifer isn't onscreen, the game never entirely lets us forget about him; our protagonist spends the entire game with a vivid reminder of their rivalry visible across his face.
Know Thine Enemy
Even though the game opens with Squall and Seifer cutting each other's faces open, there are some interesting hints of closeness between them. Seifer isn't a person who likes to show vulnerability, but, before the field exam, he looks over at Squall and asks, 'Huh? You scared, too?'
This closeness, between two characters who struggle to get close to anyone, continues to manifest throughout the exam. When Seifer decides to go against orders and Squall backs him up, Seifer seems touched by Squall's support:
Squall: I stand by the captain's decision.
Seifer: ...Captain's decision? (putting a hand on Squall's shoulder) You want to wreak some havoc too, don't you?
Squall: (shrugging off Seifer's hand) It's a good opportunity to test out my training. Thanks to you, I feel like I can take on anyone. Even if they do fight dirty, like you.
Seifer: You'll thank me when the time comes.
Zell: What the hell... I thought you guys didn't get along? You're like, all buddy-buddy now.
Before entering the communications tower, Seifer asks Squall a series of questions, about his fears and dreams, that suggest he wants to understand Squall better. Squall, in an unusual moment of showing interest in other people, asks about Seifer's dream in return:
Seifer: This must be your first real battle. You scared?
Squall: ...I don't know. I try not to think about it.
Seifer: I love battles. I fear nothing. The way I look at it, as long as you make it out of battle alive, you're one step closer to fulfilling your dream.
Squall: What? Your dream?
Seifer: You have one too, don't you?
Squall: ...Sorry, but I'm gonna pass on that subject.
'...I don't know. I try not to think about it' is also a surprisingly sincere response from Squall when Seifer asks if he's scared. If the question came from anyone else, I would expect Squall to remain silent or brush it off.
When Squall and Zell pass the SeeD exam, and Seifer fails yet again, it's an interesting detail that, after a moment's pause, Seifer is the first to start applauding. Seifer is proud, petty and short-tempered, but he's prepared to acknowledge his rival's success over him.
Even when Seifer isn't present, Squall demonstrates an understanding of Seifer in his thoughts. He'll occasionally think of how Seifer would react to something:
Rinoa: How can you be so casual? I feel... sorry for him.
Squall: (...Feel sorry? Seifer would hate to hear that.)
Rinoa: Zell wants your support.
Squall: (I knew it was gonna be something like that...)
Rinoa: Any kind of encouragement will make...
Squall (That's just to ease your mind. Am I the only one who thinks that? No, I'm sure Seifer...)
Squall has a hard time understanding other people, so it feels significant that he knows Seifer so well.
There's a respect and understanding between Squall and Seifer that, early in the game, is hard to find in their relationships with anyone else. When the game begins, I don't think Squall thinks of himself as close to anyone, but I'd say Seifer is the person he's closest to.
Squall can't allow himself to form connections with other people; that creates the unacceptable risk that he'll lose them. But Seifer slipped through his defences and connected with him by being antagonistic, rather than friendly, meaning Squall didn't feel compelled to push him away.
The Sorceress's Knight
Later in the game, when they're outright enemies, Seifer has become more violent and that undercurrent of mutual respect is harder to spot, but it's clear that Squall is still important to Seifer in some warped way.
During the D-District Prison sequence, Squall is the one Seifer chooses to torture first, and a lot of Seifer's lines in the torture scene feel very telling:
'I was hoping you'd be there, Squall. So... how'd I look in my moment of triumph?'
'This is the scene where you swear your undying hatred for me!'
'I'll be Edea's bloodhound and hunt down every one of your kind. It'll be fun, Squall. Don't die on me yet.'
He doesn't just want to be the sorceress's knight; he wants Squall to see him being the sorceress's knight. He wants to know that Squall is wounded by his actions. Being perceived by Squall is a crucial part of his dream.
Much later, when Seifer has abducted Rinoa and is trying to hand her over to the sorceress Adel, we get this exchange:
Rinoa: ...Seifer. No more... Please?
Seifer: Rinoa...
Squall: (arriving on the scene) Seifer! NO!
Seifer: Rinoa and Adel! The sorceresses as one! Watch closely, Squall! (shoves Rinoa at Adel)
When Rinoa appeals to Seifer, Seifer hesitates. But then Squall turns up, and Seifer immediately goes through with his plan. He can't show hesitation or weakness in front of Squall; he has a very firm idea of how he wants Squall to see him.
What this suggests to me is that, after everything, Seifer still respects Squall. He cares about what Squall thinks of him, and Seifer isn't someone who cares about the opinions of people he doesn't respect.
Unfortunately for Seifer, he wants Squall to hate him, and I don't think Squall does. Squall's not a person who really hates others or even views them as evil; as far as he's concerned, everyone is just fighting their own battles for their own reasons. Seifer's a person with very passionate feelings, and I suspect it frustrates him that he so desperately wants to get some sort of passion and anger from the more muted Squall.
It's a shame that we never really get closure on Seifer and Squall's relationship. Seifer handing Rinoa over to Adel is the last scene they have together. I suppose that's what fanfiction is for.