It's about time we had a spherical chess variant, and the 'GeoBoards' are certainly among the most natural choices. My intuition isn't good enough to tell me how playable this game is, but it would be fun to try.
There are several points where the rules need clarification:
What is the definition of 'forward' for Pawns and Squires? The definition that seems most natural doesn't satisfy the statement that a Pawn has three forward directions to choose from.
Regarding promotion, what exactly is meant by 'any higher-ranking piece'? My guess is Queen, Rook, or Templar. Certainly promotion to a second King would change the game drastically. But can a Pawn promote to Squire? Can a Pawn or Squire promote to Obelisk? If so, how does that Obelisk behave, given that it is outside of its Home Territory?
Is promotion mandatory when a Pawn or Squire enters the opponent's Home Territory? If not, is it mandatory when it reaches the opposing King's starting hex?
How does a Templar move? There seem to be two contradictory statements: 'The Templar is a combination of the traditional Knight and Bishop.... The Templar moves three spaces diagonally, or two spaces forward and one to the side.' Which of these is correct?
Can a Queen, Rook, or (maybe) Templar make a 'null move', traveling all the way around the board and returning to its starting space?
Is the King really in Exile only when forced out of its Home Territory?
And a comment: I tend to be suspicious of games that let Pawns take multiple steps but don't allow en passant capture. But it's possible that the geometry of this game is different enough from Chess that this won't be a problem. Does anyone have thoughts about this?
There are several points where the rules need clarification: