I am of the belief I never really saw the left half of people's faces. The term for this is called pseudoneglect and it is common for a right hander to pay more attention to the left half of his her visual field. (which would be the right side of someone else's face) There are many studies showing that seeing and mimicking of others face's recreates the emotions and muscular actions in ourselves. I believe the seeing of the left half of other people's face now and the persona reflected in it allows for a greater differentiation of what I do with my left and right half of face.
If I am correct the the lateralization of the brain for speech and right handedness lies in the lack of differentiation of both perceiving others and myself. It still remains the great possibility that there are both structural and developmental aspects of the nervous system that starts and kept me on the path of not perceiving the differences.
Faces and emotion