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