When you look at someone in the eye, you can really only look into one of their eyes at a time (or it's very difficult to look into both at once).
My right eye is dominant. I usually look straight across, my right eye into their left. Sometimes I'll glance over into their left.
EDIT: Eye dominance test.
So how do you look someone in the eye?
Do you look across, or over, or both? Which one first?