Solution:

This solution has a 4-fold rotational symmetry. One can wiggle it around to get an infinite family of solutions with the same 4-fold symmetry. For example one could introduce a little bump in an edge, provided you also introduce three other symmetrically positioned bumps. This leads naturally to:

Question: Does every solution have a 4-fold rotational symmetry?

Note: if we drop the hypothesis that the regions are connected, there is a solution with (merely) bilateral symmetry.