  The decays $K_L \to \gamma\gamma$ and $K_L \to \ell^+ \ell^- \gamma$ are
studied at the leading order $p^6$ in Chiral Perturbation Theory. One-loop
contributions stemming from the odd intrinsic parity $|\Delta S| = 1$ effective
Lagrangian of order $p^4$ are included and shown to be of possible relevance.
They affect the decay $K_L \to \gamma\gamma$ adding to the usual pole terms a
piece free of counterterm uncertainties. In the case of the $K_L \to \ell^+
\ell^- \gamma$ decays the dependence of the form factor on the dilepton
invariant mass requires a counterterm. The form factor may receive a sizeable
contribution from chiral logarithms. Including considerations from the $K_L \to
\pi^+ \pi^- \gamma$ direct emission amplitude, we obtain two consistent
scenarios. In one scenario the long distance contributions from the one-loop
terms are important, while in the other they are marginal. In both cases the
counterterm is shown to be significant.
