  We construct hadronic amplitudes for the three-body decays
$\eta^{(\prime)}\to\pi^+\pi^-\pi^0$ and $\eta'\to\eta\pi^+\pi^-$ in a
non-perturbative fashion, allowing for $C$- and $CP$-violating asymmetries in
the $\pi^+\pi^-$ distributions. These amplitudes are consistent with the
constraints of analyticity and unitarity. We find that the currently most
accurate Dalitz-plot distributions taken by the KLOE-2 and BESIII
collaborations confine the patterns of these asymmetries to a relative per
mille level. Our dispersive representation allows us to extract the individual
coupling strengths of the $C$- and $CP$-violating contributions arising from
effective isoscalar and isotensor operators in
$\eta^{(\prime)}\to\pi^+\pi^-\pi^0$ and an effective isovector operator in
$\eta'\to\eta\pi^+\pi^-$, while the strongly different sensitivities to these
operators can be understood from chiral power counting arguments.
