To make a move, you would rotate a column or row.
To rotate a column/row, you would shift the numbers over one position and wrap around a number from one end to the other while flipping its value from zero to one and vice versa.
The goal is to end up with the same number in all 9x9 locations.
What do you think of this idea?
For a physical puzzle, it would be a fun object, but I don’t see how to build it.
(Topologically, it’s a torus with a 9 × 9 grid, where each 9-item ‘row’ and ‘column’ is a Möbius strip with a zero on one side of each item and a one on the other (I would use two colors to make it more attractive))