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Ah, that reminds me of another thing I was thinking about.  Start with a
triangular board, then consider playing on the edges.  Each edge has six
adjacent edges, but two of them lie along the same line as the given edge
(and are, under the Euclidean metric, further away).  So we have two
reasonable ways to play.  We can literally treat all adjacent edges as
'orthogonal' moves, which I think should turn the game into fairly
standard hex movement.  Or we could exclude these two funny edges, perhaps
making them into a new type of move.  Then we have a hex game which singles
out certain kinds of orthogonal moves as special (but these special
directions don't seem to be universal; without a drawing I'm having
trouble seeing how they work together...)