The answer lies in the fact that in the early days of chess (when the queen wasn't even the queen) squares had no colors, but were just separated by lines. It was then natural to arrange all pieces such that they face their counterparts.
Historical significance of "queen on her color"
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Just wondering if any chess historians out there know the answer to this one. Why have players over the years settled on the current board arrangement, rather than the obvious:
Is there some kind of problem here I'm not seeing?