  We derive constraints on the four dimensional energy-momentum tensor from
gravitational and gauge anomalies. Our work can be considered an extension of
Duff's analysis [1] to include parity-odd terms and explicit symmetry breaking.
The constraints imply the absence of the parity-odd $R\tilde R$ and $F\tilde F$
terms, for theories whose symmetries are compatible with dimensional
regularisation, in a model-independent way. Remarkably, even in the case of
explicit symmetry breaking the $\Box R$-anomaly is found to be finite and
unambiguous after applying the symmetry constraints. We compute mixed
gravity-gauge anomalies at leading order and deduce phenomenological
consequences for vector bosons associated with global chiral symmetries.
