Decays of the neutral and long-lived $\eta$ and $\eta'$ mesons provide a unique, flavor-conserving laboratory to test low-energy Quantum Chromodynamics and search for physics beyond the Standard Model. In these proceedings, we specifically concentrate on discrete symmetry tests. We point out how stringent constraints from electric dipole moment measurements strongly limit the realistic opportunities to observe $P$- and $CP$-odd mechanisms in $\eta$ or $\eta'$ decays. For the far less investigated $C$- and $CP$-odd processes, we introduce a dispersion-theoretical formalism to assess these from Dalitz plot asymmetries in a model-independent way.