Doug Lindstrom is currently developing a paper on totally antisymmetric torsion. It comes out that the torsion tensor then is homomorphic to a 4-vector. This will give new insights, including a new reduction of polarization indices.
I myself tried out tetrad matrix examples which have diagonal form. This means that the coordinate systems of the base manifold and tangent space are collinear. In terms of Cartan geometry, this means that there is a 1-to-1 correspondence between Greek and Latin indices. As a next step, I will develop the e-m field from torsion (which is simply a re-numbering of torsion tensor elements). The tetrad is the e-m potential, and a diagonal tetrad means that there is exactly one scalar potential and one vector potential. We will see if meaningful results will follow from this approach.