Actually, the energy momentum tensor (EMT) does not have to be symmetric. If the metric tensor is not symmetric then neither is the EMT. Einstein tried to relate the non-symmetric part to the electromagnetic field, but this never panned out. Recently it has been shown that the antisymmetric part...
If the metric tensor is not symmetric then the energy momentum tensor is not symmetric, and so, we have the physical interpretaion that the antisymmetric part of the metric tensor is related to spin. In fact, it is the potential of the spin field (torsion). The mathematics is given in...