Motivated by the search for possible CP violating terms in the trace of the
energy-momentum tensor in theories coupled to gravity we revisit the problem of trace
anomalies in chiral theories. We recalculate the latter and ascertain that in the trace of
the energy-momentum tensor of theories with chiral fermions at one-loop the Pontryagin
density appears with an imaginary coefficient. We argue that this may break unitarity, in
which case the trace anomaly has to be used as a selective criterion for theories, analogous
to the chiral anomalies in gauge theories. We analyze some remarkable consequences of
this fact, that seem to have been overlooked in the literature