  We revisit well-known short-distance constraints relating the hadronic
light-by light Green's function to the $\langle VVA \rangle$ one, that have
been a subject of debate over the past years in the context of the muon
$(g-2)$. Specifically, we identify a relation among the longitudinal and
transverse degrees of freedom that is enforced by the axial anomaly that, by
contrast, has not received attention in the past. Such relation allows, among
other things, to overcome the problem of basis ambiguities when describing
axial-vector mesons transition form factors, but further applications are
discussed as well, with special focus on the role of axial-vector mesons in the
HLbL contribution to the muon $(g-2)$. Our results should also contribute to a
better understanding of the, so far, controversial interplay among
short-distance constraints with longitudinal and transverse degrees of freedom,
such as axial-vector mesons. This is key to confront the theoretical and
experimental result for the muon $(g-2)$ that, currently, exhibits a
$4.2\sigma$ tension.
