The basic result is the
classification of first order invariant elliptic differential operators on a quotient of a
spin symmetric space by a suitable discrete group: Such operators are all twisted
Dirac operators. As a consequence we obtain conditions for the spectral symmetry to
be equivariant. We also show that the characteristic numbers of these spaces vanish,
a result previously obtained by Hirzebruch and Slodowy from the study of elliptic
genera.