I believe it has to do with brain lateralization. The asymmetry would not be the cause, but is merely the way the modern human brain functions. The lateralization is the older characteristic, and made asymmetry possible.
20+ years ago there was a book, "The Origin of Consciousness in the Breakdown of the Bicameral Mind" that is pertinent to this discussion, and Tim what's his name probably is familiar with it.
The Origin of Consciousness
in the Breakdown of
the Bicameral Mind
by Julian Jaynes