  Isospin-breaking (IB) effects are required for an evaluation of hadronic
vacuum polarization at subpercent precision. While the dominant contributions
arise from the $e^+e^-\to\pi^+\pi^-$ channel, also IB in the subleading
channels can become relevant for a detailed understanding, e.g., of the
comparison to lattice QCD. Here, we provide such an analysis for $e^+e^-\to
3\pi$ by extending our dispersive description of the process, including
estimates of final-state radiation (FSR) and $\rho$-$\omega$ mixing. In
particular, we develop a formalism to capture the leading infrared-enhanced
effects in terms of a correction factor $\eta_{3\pi}$ that generalizes the
analog treatment of virtual and final-state photons in the $2\pi$ case. The
global fit to the $e^+e^-\to 3\pi$ data base, subject to constraints from
analyticity, unitarity, and the chiral anomaly, gives $a_\mu^{3\pi}|_{\leq
1.8\,\text{GeV}}=45.91(53)\times 10^{-10}$ for the total $3\pi$ contribution to
the anomalous magnetic moment of the muon, of which
$a_\mu^\text{FSR}[3\pi]=0.51(1)\times 10^{-10}$ and
$a_\mu^{\rho\text{-}\omega}[3\pi]=-2.68(70)\times 10^{-10}$ can be ascribed to
IB. We argue that the resulting cancellation with $\rho$-$\omega$ mixing in
$e^+e^-\to 2\pi$ can be understood from a narrow-resonance picture, and provide
updated values for the vacuum-polarization-subtracted vector-meson parameters
$M_\omega=782.70(3)\,\text{MeV}$, $M_\phi=1019.21(2)\,\text{MeV}$,
$\Gamma_\omega=8.71(3)\,\text{MeV}$, and $\Gamma_\phi=4.27(1)\,\text{MeV}$.
