We derive two types of dispersion relations for the hadronic vector-vector-axial-vector (VVA) correlator: one in generic three-point kinematics with fixed photon virtualities, the second in the kinematic limit of one soft photon. The VVA correlator enters in the electroweak contribution to the muon anomalous magnetic moment and it also emerges as the leading term in the operator-product expansion for hadronic light-by-light scattering in the limit of two large photon momenta. Previously, this correlator was only described in terms of hadronic models. Our new dispersive treatments are in analogy to the established and the newly proposed dispersive approaches to hadronic light-by-light. The VVA correlator allows us to investigate the relation between the two types of dispersion relations in a simpler example and to elucidate the reshuffling of different hadronic contributions in this comparison. As a byproduct, we reduce the theoretical uncertainties on the first-family VVA contribution to the muon anomalous magnetic moment by combining the dispersive representation with both asymptotic and low-energy constraints.