Shinji is straight. He is attracted to females. Lots of females, because he is a fourteen-year-old boy and he is surrounded by attractive members of the opposite sex who wear skintight clothing. Or in Misato's case, really short skirts. But his emotional connections to most of them are tenuous at best. Although he does form a bond with Misato, he also places limits on it, often avoiding her offers of comfort or help. He and Rei are friends in a way, as much as her emotional distance will allow, and until the later parts of the series, I considered her the most plausible romantic partner for Shinji. (Yes, it's incestuous. I didn't say I STILL think that.) However, his relationship with Asuka is competitive, prickly, and sometimes downright hostile. Her verbal barbs lodge deeper in him than she probably realizes.
I said that a relationship with Rei would be incestuous. I don't know that it's ever explicitly stated in the series (I saw the last few volumes fansubbed... not a fun experience) but Rei is a clone of Shinji's mother. Misato is another mother figure, and that relationship deserves an essay in its own right. Suffice it to say that he often seems to avoid Misato's touch or comfort as a way of avoiding the sexual implications of closeness with her. Asuka becomes, by the end of the series, the only option.
Their "relationship" seems founded on very little; Shinji is attracted to her sexually, and Asuka dances around him, taunting and inviting, trying to play a game she doesn't understand. She uses him to have a first kiss, then pushes him away furiously. By the beginning of the movie, he clings to her the way a baby monkey clings to a wire "mother" -- she may not be much, but she's all he has left anymore. Kaworu is gone, no one else offers the sympathy he needs, so he's left to masturbate while she lies unconscious. Antiseptic, emotionless, solitary sexuality, desperation, and habitual, almost meaningless insults, are the nature of Shinji's relationship with Asuka.
Kaworu can't help but come off well in comparison. The defenses that are at their strongest around Asuka are dropped when Shinji is with Kaworu. He is able to open up to the other boy as he can to no one else in the series, and he feels safe with him. Even after Kaworu reveals himself as the angel, which Shinji views as a betrayal, his feelings remain. In the movie, Shinji is confronted by a giant version of Rei, who has merged with Lilith. He's terrified, screaming... until the Rei-monster transforms into Kaworu. Shinji calms almost instantly, his expression peaceful and happy as he calls out to Kaworu-kun.
The latter part of End of Evangelion is the clearest to me both in meaning and in my memory of it. It recaps the "Human Instrumentality" part of the end of the TV series, with a stronger focus on Rei and especially Asuka. This, no doubt, is where the "soulmate" belief comes from, as the two spend much of the time shouting back and forth, arguing, disagreeing. Not in concord. In the end, Shinji must choose whether or not to accept separation from others, disunity, pain... and he does. The movie ends with Shinji and Asuka, together. Shinji tries to choke her, Asuka caresses his face lightly and he releases her, and she complains that she feels sick.
Shinji has made the decision for individuality, separation, and this is the consequence. It's difficult to imagine anyone Shinji is MORE separate from than Asuka. Unlike Kaworu, she is in opposition to Shinji, emotionally, mentally, and on the most basic level; she's the opposite sex. Shinji could be "one" with Kaworu, who's like him, while he and Asuka are a union of opposites. He could be with no one else, not without a complete change in the meaning of the entire story.