At low energies hadronic vacuum polarization (HVP) is strongly dominated by two-pion intermediate states, which are responsible for about $70%$ of the HVP contribution to the anomalous magnetic moment of the muon, $a_mu^text{HVP}$. Lattice-QCD evaluations of the latter indicate that it might be larger than calculated dispersively on the basis of $e^+e^-totext{hadrons}$ data, at a level which would contest the long-standing discrepancy with the $a_mu$ measurement. In this Letter we study to which extent this $2pi$ contribution can be modified without, at the same time, producing a conflict elsewhere in low-energy hadron phenomenology. To this end we consider a dispersive representation of the $e^+e^- to 2pi$ process and study the correlations which thereby emerge between $a_mu^text{HVP}$, the hadronic running of the fine-structure constant, the $P$-wave $pipi$ phase shift, and the charge radius of the pion. Inelastic effects play an important role, despite being constrained by the Eidelman-Lukaszuk bound. We identify scenarios in which $a_mu^text{HVP}$ can be altered substantially, driven by changes in the phase shift and/or the inelastic contribution, and illustrate the ensuing changes in the $e^+e^-to 2pi$ cross section. In the combined scenario, which minimizes the effect in the cross section, a uniform shift around $4%$ is required. At the same time both the analytic continuation into the space-like region and the pion charge radius are affected at a level that could be probed in future lattice-QCD calculations.
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