Signature reversal invariance

We consider the signature reversing transformation of the metric tensor g_ab goes to -g_ab induced by the chiral transformation of the curved space gamma matrices gamma_a goes to gamma gamma_a in spacetimes with signature (S,T), which also induces a (-1)^T spacetime orientation reversal. We conclude: (1) It is a symmetry only for chiral theories with S-T= 4k, with k integer. (2) Yang-Mills theories require dimensions D=4k with T even for which even rank antisymmentric tensor field strengths and mass terms are also allowed. For example, D=10 super Yang-Mills is ruled out. (3) Gravititational theories require dimensions D=4k+2 with T odd, for which the symmetry is preserved by coupling to odd rank field strengths. In D=10, for example, it is a symmetry of N=1 and Type IIB supergravity but not Type IIA. A cosmological term and also mass terms are forbidden but non-minimal R phi^2 coupling is permitted. (4) Spontaneous compactification from D=4k+2 leads to interesting but different symmetries in lower dimensions such as D=4, so Yang-Mills terms, Kaluza-Klein masses and a cosmological constant may then appear. As a well-known example, IIB permits AdS_5 x S^5.