On manifolds with non-trivial Killing tensors admitting a square root of the Killing-Yano type one can construct non-standard Dirac operators, which differ from, but commute with, the standard Dirac operator. We relate the index problem for the non-standard Dirac operator to that of the standard Dirac operator. This necessitates a study of manifolds with torsion and boundary and we summarize recent results obtained for such manifolds.