  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 $e^+e^-\to\text{hadrons}$ 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 $2\pi$ 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 $e^+e^- \to 2\pi$ process and study the correlations
which thereby emerge between $a_\mu^\text{HVP}$, the hadronic running of the
fine-structure constant, the $P$-wave $\pi\pi$ 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 $e^+e^-\to 2\pi$ 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.
