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The classical formulation:
The theorem shouldn't be expected to work if the image is decomposed into a cubical complexes without change either. The reason is this. The realization of a cubical complex in the plane is a closed set, hence its complement is open, so it's not a cubical complex.
This is a version of Jordan theorem for cubical complexes (similar for cell complexes):
The complement of a closed curve $C$ with no self-intersections as a $1$-dimensional cubical complex in the plane is the union of two disjoint sets, the closures of which are connected cubical complexes $A$ and $B$ with intersection equal to $C$.