I was doodling with a 17cm square, divided into an 8×8 grid and collapsed, via a photodiagram (and a bit of wrestle-magic) into a curious but possibly useful surface corrugation/tessellation:

With an exercise in patience, fold accuracy and layer management, a “swastika”-like collapse becomes a sunken 4 segment recess, then the edges tidy up with some propagatible pleats, making this tesselatable.

Folding this fairly big (grid here is 1.5cm) was mind-bending but manageable. The challenge will be to try this on a smaller grid, integrated into a larger structure.