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I'd been thinking about this particular question for a little while, and
happened to run across this, unanswered, from 2002:
2002-07-07	SBlkWlf Unverified	None	
i'm just wondering why in most multi-level chesses the boards alternate
their patterns (instead of a constant white in the bottom right), and this
is the only place i could think to ask.  Is it necessary for some reason? 
Would colorbound pieces be affected adversely if the boards weren't so
arranged ?
Anyone...anyone...

The answer is that it depends on the exact movement patterns of your pieces
through the levels. For the most common extension of pieces from 2D to 3D
and 4D, the reverse checkering is required. However, when I designed my
first chess variant, Hyperchess, I didn't realize this, so I made every
level the same pattern of light and dark. For the simplified 4D movement
which Hyperchess uses, this is a much better board design. I suspect that
it's possible, even likely, that how you checker your higher-dimensional
board levels will affect your game design on that board. The different
patterns lend themselves to different ways of looking at how to design
piece moves.