[yaatsxzd] Rotating a Tetris piece

Tetris piece rotation could be much more sophisticated than typically done.

Choose the center of any square or any corner to be the center of rotation by 90 degrees, or the midpoint of any edge for rotation by 180.  With this much freedom, you can walk a piece upwards by successively choosing different centers of rotation, so maybe don't fight it and simply provide up as a valid direction.  The game will have to change, maybe playing only against a clock or limited total number of pieces.  Or, we could constrain only to rotations that do not raise the center of mass, keeping with the theme of Tetris that blocks only fall.

Does only the final orientation need to be clear of blocks or does the entire swath of space consumed during the rotation?  If the former, what about 180 rotations which have to pass through a blocked 90 rotation?

If the latter, it feels like threading a piece through holes.  More rotations could probably be made by making squares slightly smaller within the grid and rounding off corners.  The ultimate would be a skeleton of line segments connecting square centers.  Some rotations might only be possible when the piece starts not aligned with the grid.

Allowing simultaneous rotation and translation reminds us of the moving sofa problem.  Are there interesting shapes made of blocks for which simultaneous rotation and translation allow movements not possible with rotation and translation separately?