Using a sample of 2.59 times 10^7 psi(2S) decays collected by the CLEO--c detector, we present results of a study of chi_{c0} and chi_{c2} decays into two-meson final states. We present the worlds most precise measurements of the chi_{cJ,(J=0,2)} -> pi^+pi^-, pi^0 pi^0, K^+K^-, K^0_S K^0_S, eta eta and eta^{prime} eta^{prime} branching fractions, and a search for chi_c decays into eta eta^{prime}. These results shed light on the mechanism of charmonium decays into pseudoscalar mesons.