We have recently shown that the large direct CP violation observed in the decay B0→π+π−, and the enhanced branching ratio for B0→π0π0, can be understood by invoking a small mixing of the ππ system with the dominant DD¯ channel. We examine here the analogous effect of DD¯↔KK¯ mixing on the rare decays B0→K0K¯0, B−→K−K0 and B0→K+K−. We find (a) significant values for the asymmetry parameters C and S in B0/B¯0→K0K¯0, (b) a possible enhancement of the suppressed mode B0→K+K−, (c) a correlation between the three decay channels following from a triangle relation between amplitudes AK0K¯0−AK+K−=AK−K0. The pattern of asymmetries and branching ratios is compared with that derived from the short-distance QCD penguin interaction.