For instance, "pawn odds" is really "pawn and move" odds. That is, the odds giver plays Black and removes his KBP.
Why not allow an "intermediate" odds where the odds giver plays white? And why not allow the odds give to choose which pawn to remove?
In the case of knight and rook odds, it is the queen knight or rook that's removed, but why not allow the odds giver to substitute the king's pieces instead?