If we just look at the exterior surface of this, and view it as a 2-manifold, is this the direct sum of two Mobius bands? I see that this object has 2 sides and is orientable. If we view it as a 3-manifold, is that what makes it a direct sum of two Mobius bands, since we have $S^1\times \mathbb{R}^2$ in some way?
If I try to consider Euler characteristics, I get 11*\chi(cube). I think that’s just the statement that the cubes don’t give a triangulation of M x M, so we don’t get \chi(M x M) = \chi(M)•\chi(M)=0•0=0 for M a Moebius band. However, shouldn’t we still be able to view the cubes as gluing together to give a CW complex?