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Contrary to what some have written, this game is indeed spherical. Firstly,
the right and left edges are connected to each other, making a cylinder (or
annulus for the mathematically pedantic). Then imagine shrinking the top
and bottom circles on this cylinder until they become mere points. This
means that the top eight 'squares' are really triangles, all joining at
the top edge of the board, which is now a point. The same goes for the
bottom. The black pieces are then in a circle around the north pole and
the white ones around the south pole. For examples of what this looks
like, see:

http://69.90.174.250/photos/thumb_small/69461/69461,1165605152,1.jpg
http://www.cs.unc.edu/~olano/papers/primitive/sphere_check.gif
http://upload.wikimedia.org/wikipedia/commons/b/b3/UV_mapping_checkered_sphere.png

(although they all have too many squares, it should be eight from pole to
pole and eight around the equator)

For reference, the ranks become lines of latitude and the files lines of
longitude. A rook travelling up a file comes back down the 'opposite'
file four squares to the side. The other pieces are harder to work out,
but I assure you that this entry has got the geometry correct.

I independently invented the rules about five years ago and made my own
board out of a 10cm diameter polystyrene ball, with pins as pieces. This
worked quite well and only took a few hours to make with a black marker
and a piece of string for measurements. Before making it, I determined the
rules by drawing the board as a series of concentric circles (roughly like
circular chess, but 8x8 not 4x16), then make them go all the way into the
center point, which is one of the poles. This can give a playable flat
board for the game, and help you see how going through the poles works
(although it is pretty poor for helping with the colour that begins around
the outside of the circle).

As to the game, it is quite similar to cylindrical chess. Indeed, it is
identical until you go through a pole, which cannot happen until at least
a few pieces leave the home squares. The end is quite different as the
poles become vacated and are convenient for travel.

My only rules quibble would be that you don't need to modify the castling
rule. Indeed, I would play this and cylindrical chess without any castling
as the entire point of it is void when there are no corners of the board,
and it would never have been invented on such a board.