I came up with an (as far as I know) original puzzle idea. I'll describe it in terms of how to modify the Equator
First, glue 3/4 of the puzzle in place. Two opposite tiles are left unglued, as are all tiles on two adjacent arcs connecting them, and the 1/4 arc segment connecting those two. Other than that wedge though, everything else is cemented in place.
You'll probably notice that the puzzle is now immovable. To fix that, remove a single tile, and it's now a sliding block puzzle.
This puzzle presents some unique challenges, and is very hard for a sliding block puzzle. Unlike other sliding block puzzles, its tiles have orientation, and the freedom of motion is restricted enough to make the puzzle difficult.
This puzzle could be built as just a wedge with no sphere attached. I think that would be a nice-looking and fun puzzle.