We present a summary of a model-independent, non-perturbative construction of the hadronic amplitudes
$\eta\to\pi^+\pi^-\pi^0$ and $\eta'\to\eta\pi^+\pi^-$,
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 and per cent level, respectively.
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 $\eta'\to\eta\pi^+\pi^-$.