We have previously analysed the chiral predictions for the decay KL ! 0 e+e− and discussed how experimental measurements of this process can help our understanding of chiral theories and of CP tests. Motivated by the possibility that the analogous muonic process may soon be measurable, we extend this calculation to the decay KL ! 0 μ+μ−. Branching ratios and differential branching ratios are calculated and presented. Measurements of these will help shed some light on the puzzles in our understanding of the chiral loop effects which were raised by the rate for KL ! 0 .