  We show that the sigma models with target spaces supersymmetric heterotic
backgrounds with $SU(2)$ and $SU(3)$ holonomy are invariant under a W-symmetry
algebra generated by the covariantly constant forms of these backgrounds. The
closure of the W-algebra requires additional generators which we identify. We
prove that the chiral anomalies of all these symmetries are consistent at
one-loop in perturbation theory. We also demonstrate that these anomalies
cancel at the same loop level either by adding suitable finite local
counterterms in the sigma model effective action or by assuming a plausible
quantum correction to the transformations. Such a correction is compatible with
both the cancellation mechanism for spacetime frame rotation and gauge
transformation anomalies, and the correction to the heterotic supergravity up
to two loops in the sigma model perturbation theory.
