[en] A two-component model of the decay K_L→π⁰γγ, in which the amplitude is the sum of a pion-loop contribution and a vector-meson-mediated term, is compared with data from the NA31 and E731 experiments. An estimate is obtained for the parameter G characterizing the strength of the vector-meson component. The amplitude of K_L→π⁰γγ, determined in this way, is used to calculate the CP-conserving amplitude of the decays K_L→π⁰e+e- and K_L→π⁰μ⁺μ^-, without neglect of lepton masses. This complex amplitude (containing both absorptive and dispersive parts) is allowed to interfere with the CP-violating amplitude expected from the short-distance interaction s bar d→l⁺l- and from the CP impurity of the K_L wave function. Results are obtained for the branching ratios of K_L→π⁰l+l- and for the CP-violating l⁺l- asymmetry in the Dalitz plot