Mine cycles through the no eye contact. I have noticed that when he really wants me to believe him and his lies, he stares me in the eyes intently. It is like a stare down. At first it unnerved me. It still does to some extent but I am getting used to staring back just as intently. I have moments of where he can talk to me normally and employ eye contact or I have moments where he cannot look at all in my direction. Those moments are clues that he is up to no good or the running away is starting again.
I truly believe it is a combination of shame/guilt and the fact that he knows that if he looks me in the eye, he will see "me" not the evil person he has made me out to be or maybe he is worried about what I will see in his eyes. At the beginning, there was nothing to see in those eyes. They were just dark.