  We extend a previous computation of the TJJ correlator, involving the
energy-momentum tensor of an abelian gauge theory and two vector currents, to
the case of mixed axial-vector/vector currents. The study is performed in
analogy to the case of the $AVV$ vertex for the chiral anomaly. We derive the
general structure of the anomalous Ward identities and provide explicit tests
of their consistency using Dimensional Reduction. Mixed massive correlators of
the form $TJ_V J_A$ are shown to vanish both by Ward identities and by
C-invariance. The result is characterized by the appearance of massless scalar
degrees of freedom in the coupling of chiral and vector theories to gravity,
affecting both the soft and the ultraviolet region of the vertex. This is in
agreement with previous studies of the effective action of gauge and conformal
anomalies in QED and QCD.
