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.Comment: LaTex, 31 pages; v3: Extended discussion of fermions without
vielbeins. Version to appear in Nucl. Phys.
