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We have theoretically and experimentally investigated dual-core photonic bandgap fibers (PBGFs), which consist of a cladding with an array of high-index rods and two cores formed by omitting two nearby rods. We find novel features in their coupling characteristics such as maxima and minima in coupling length, complete decoupling of the cores, and an inversion of the usual ordering of supermodes so that the odd supermode has the higher propagation constant. This behavior is understood by considering the field distribution in the rods between the cores.