  By analyzing existing data on pseudoscalar charmonium decays, we obtain the
ratio of the branching fractions of $\eta_c(2S)$ and $\eta_c$ decays into ten
different final states with light hadrons. For the first time, we test the two
existing theoretical predictions of these decays, and find that the
experimental data are significantly different from both of them. The lack of
observation of any decay mode with higher rate in $\eta_c(2S)$ than in $\eta_c$
decays suggests very unusual decay dynamics in pseudoscalar charmonium decays
to be identified. We also report the first model-independent evaluation of the
partial width of $\eta_c(2S)\to \gamma\gamma$ ($2.21_{-0.64}^{+0.88}$ keV) and
improved determination of that of $\eta_c\to \gamma\gamma$
($5.43_{-0.38}^{+0.41}$ keV). The latter shows a tension with the most recent
lattice QCD calculation.
