First in a series of in-depth MBTI character analyses.
I have often referred to INTJ as the "stereotypical scheming villain", though all the same it doesn't mean this type is naturally a villain. I think any type can reasonably find some way to become/express evil (The villain in my novels is ENTP). Anyway, to analyze the letters further, I didn't really have to think about this too much, as Ayanami is more or less a model example of INTJ. Scheming appears to be one of the things that Ayanami does best. He comes up with some pretty elaborate, unexpected things, some of which seem even too bold for his type, but that is easily explained. INTJ is a type that can certainly come off as arrogant because of their amazing self-confidence. That's something we can see just radiating from his person. He's obviously very confident in any decision he makes.
He is one of the most introverted villains I have seen in a long time. Most villains have to be extroverted to some extent in order to obtain and keep followers. Ayanami just got lucky in the case, it seems, because he did not have to obtain his followers, but merely keep their loyalty. Not to say that he didn't work his way into the system stealthily.
Intuitively, Ayanami is extremely gifted. His dominant function is introverted intuition, which means that more often than not the first thing he does is intuiting. This means that he starts linking things together; working them out, testing and retesting. I can relate to this as my personal type also has introverted intuition as a dominant function. You can see this in the character by how much he seems detached from the world. Intuition isn't really "thinking" as much as it is introspecting - seeing symbols and their meanings... etc. Ayanami is always planning, "scheming" so to say. However, Ayanami also exercises his extroverted Sensing very well. He wouldn't be able to fight appropriately without that ability, and would probably feel at a loss. He also uses his Sensing for other things, and it is primarily the source of his social interaction.
It pretty much goes without saying that Ayanami is a Thinking type, and pretty much solely so. I can't think of a single thing he has done that he has done for his own emotional fulfillment. Granted, he makes it obvious that he is very aware of the emotional needs and states of others (probably something he attains from his intuition) and uses that, logically, to his advantage. (If you don't know what I mean, think about what he did to Mikage. He was well aware of what emotional reaction he would evoke from Teito when he did it.) I remember reading at some point that one of the characters was wondering about Ayanami having any emotion... It was something about them wondering why he had told his forces to retreat, and they were wondering if he had some kind of emotional connection to them. Regardless of who was saying that, I disagree; I think Ayanami's reasons to retreat are logic alone. However, it is not that Ayanami has no feelings whatsoever for anyone. We all know that Verloren certainly did have feelings for a certain someone... It's evident however that this is not at all normal for Ayanami, and probably unsettles him. When Kuroyuri tells him to abandon him to protect himself, he refuses to do so. I can't say I know if this is because of emotion or logic, though it if is emotion rather than the logic he is so used to, it may be because he is stressed and his shadow functions are emerging (extroverted Feeling). It's also obvious that he cares deeply for his former Begleiter, Yukikaze. My theory is that Yukikaze was able to respect him as a person as well as for his leadership and ideas, in a way that Ayanami was not used to, but that made him feel more whole, and intrinsic to the well-being of another person. Because of this, Yukikaze became dear to him. This is similar to the relationship between Castor and Razette. There is no evidence that Ayanami was possibly less cold of a person in the past, though it might be that he became more this way after Yukikaze's death.
His judgment preference is what gives Ayanami his aura of self-confidence. Granted, the INT part of himself is easily enough for him to create plans and solution that work. An INTP could come up with things that are just as effective, however, they lack confidence in their decisions and tend to waver in their envisioned value of their work. As we all can well see, Ayanami is set in his ways, though his Judgment preference is not as strong as his introversion, intuition and thinking. We can see this in the way that he is part of a structured organization and is still bending the rules. He breaks the mold just enough to get a little bit of attention. This is the only one of his preferences that I can see being challenged, because of this. However, he is so decisive and follows through without wasting any time. Also, he seems to be immaculately organized. All of these things are Judgment traits.
Can an INTJ lead? This is an important thing to consider for Ayanami. It isn't a natural position for an INTJ, though I think it's obvious that in Ayanami's work that he is a very hands-off leader. Granted, INTJs naturally gravitate toward responsibility. Ayanami doesn't explicitly moderate everyone's actions (which is probably a good thing for Katsuragi!) and he doesn't seem to keep them under too tight of constraints. This might be another area where one could argue that he is INTP, because of how judgment types tend to organize people as well, and perceptual types typically are not decisive enough to be leaders. Ayanami seems to be more of a strategist than a leader. His skills are in working things out, not necessarily in making the orders, though he has gotten used to that. Ayanami's focus is on the future, and how it will be impacted by the intricate decisions and strategies he makes
Another interesting trait that we have seen Ayanami showing is empathy. Empathy!? What!? Empathy doesn't just come from Feeling types. Granted, Feeling can be a good medium for empathy, however I think Ayanami's comes from dominant introverted Intuition paired with tertiary extroverted Sensing. We can tell that he knows what people are thinking and feeling (again, I bring up the situation with Teito, because it is such a good example). Ayanami knows that Teito is a very emotional person, as well as the type who doesn't think before acting, that much he has understood from his actions. He formulated his plan to control him from that moment.
Does this mean that Ayanami knows that Katsuragi is a spy? Probably. Though it seems that Katsuragi is very well-trained at hiding his true intentions, Ayanami could very well deduce them from interacting with him. My thought is that he is suspicious, and is keeping an eye out and keeping Katsuragi at arm's length at the same time. We already know that Ayanami does not seem to trust him very much, and this is probably because he has insight into who is and what his aims are.
Ayanami's order of function preference is introverted Intuition, extroverted Thinking, introverted Feeling, extroverted Sensing, extroverted Intuition, introverted Thinking, extroverted Feeling, and introverted Sensing.
Ayanami is the NT - Rational temperament, which means he is technical, competent, ingenious, skeptical, innovative, and a non-conformist.
Some things that Ayanami would probably do under stress are: pursuing dead-end ideas, worrying about his social image, getting caught up in the past, ignore his own desires to fulfill those of others, fight with his emotions, ask too many internal "what if" questions, become very disagreeable, become even more difficult to interact with socially.
Ayanami is probably an Enneagram 5 wing 4.