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When playing on borders, how do you define an adjacent location?  Is it a
border that shares an endpoint with the current one?  I suspect a lot of
these ideas will wrap around onto one another (for instance, the duality
between triangular and hexagonal boards, the self-duality of rectangular
boards, etc.)  I think also that when you pass between these viewpoints,
some of the pieces may be defined differently from their standard.  (That
is, if you define what you think pieces should do in a hex corner game,
perhaps they will move differently than what their counterparts in
triangular chess usually do.)  This may or may not be a good thing.  I
generally like it, but it's nice too if they work out to be consistent.