Hi Alan,
We may be thinking along parallel lines.
If a Black Hole is static (I'm not sure that it can really exist, but consider the question), then quantum gravity may build a spacetime lattice similar to a Carbon-60 buckyball/ soccer ball/ truncated icosahedron just outside of and surrounding the singularity. This symmetry could potentially be stable against the crushing gravitational (near?) singularity.
If we have a rotating Black Hole, then torsion from the spinning Black Hole may cause a pair of nested Buckyballs to morph into their homotopic cousin, a lattice-like torus. This lattice-like torus has some similarities with your Mobius figure.
I was worried about how discrete toroidal lattices could transition into continuously differentiable spacetime, but it may be as simple as Philip Gibbs "qubits of strings" idea. Basically, the lattice point is the end of a string (and the strings expand within the Black Hole as Sreenath's logarithmic spirals), and a large number of lattice points and strings exist (10^41 or more) such that spacetime seemingly blends into a continuum.
CPT symmetry almost makes it sound as if different pitched or rotating Archimede's Screws may represent forward vs. backward time evolution, matter vs. anti-matter, and attractive gravity vs. repulsive dark energy/ Cosmological Constant.
If you read my essay, you will see that I have been trying to incorporate the different spin statistics (Maxwell, Bose and Fermi) into my ideas. I think that Inflation was caused by the breaking of the original TOE symmetry. We may generate self-similar scales via such kinds of phase transitions (as Inflation - this also ties into Coldea et al's magnetic quasi-particle masses near a phase transition), and our Observable Universe may be just one of several self-similar scales (the quantum scale is another).
Have Fun!
Dr. Cosmic Ray