Interesting one! I played maybe a half-dozen levels, up to Microbes. It might be interesting if a button swapped the 'view', rather than, or in addition to, the current action button swap of the 'movement dimension'... that is, i could imagine X button changing the whole map appearance and the player always moving inside just one of the smaller squares, if that makes sense.
This also very vaguely reminds me of 'Counter Hero', a game where you could 'swap numbers' in kind of arbitrary game mechanics; here it's kind of like the action button swaps dimensional numbers, like your x-coordinate becomes your z-coordinate or something... this game's mechanic seems like something that might 'stack' onto late-game puzzles of other sokoban-likes to create a new twist, maybe, somehow.