  Using a sample of 3 x 10^6 psi(2S) decays collected by the CLEO III and
CLEO_c detector configurations, we present results of a study of \chi_{c0} and
\chi_{c2} decays into eta-eta, eta-etaprime, and etaprime-etaprime final
states. We find B(chi_{c0} -> eta-eta)= (0.31+-0.05+-0.04+-0.02)%, B(chi_{c0}
-> eta-etaprime) <0.05 % at the 90% confidence level, and B(chi_{c0} ->
etaprime-etaprime) = (0.17 +- 0.04 +-0.02 +-0.01)%. We also present upper
limits for the decays of \chi_{c2} into these final states. These results give
information on the decay mechanism of \chi_c states into pseudoscalars.
