Binary colloidal mixtures have the potential to greatly expand the range of mesoscale structures fabricated by self-assembly, putting them at the frontier of the realization of novel photonic crystals. Currently, it is not understood how the multicomponent basis and particle size ratio of binary colloidal crystals affect the photonic dispersion relations. Zincblende structures are targets for binary colloidal assembly since they are closely related to well-known photonic crystal structures such as diamond cubic and inverse opal. In this investigation, we model zincblende with a crystal basis consisting of two spheres having independently variable radii and dielectric constants. Electromagnetic calculations implemented in the MIT Photonic Bands package (MPB) are used to compute the dispersion relations and determine the size and location of complete photonic bandgaps as a function of the structural and materials parameters. We examine both single and binary compositions, in which the dielectric constants of the constituent spheres are equal or unequal, respectively. Bandgap sizes are found up to 31.9% (gap-to-midgap ratio) in the inverse structure and 13.9% in the direct structure. These gap sizes compare favorably with those of inverse diamond (30.7%, ε = 16) and inverse opal (8.0%, ε = 16). Symmetry reduction at the lattice points of the diamond structure (via a basis of spheres of different size) opens a bandgap between the fifth and sixth bands in the direct structure. This gap has only previously been reported in a handful of related structures. Our theoretical modeling points experimentalists in colloidal synthesis and assembly toward these high payoff target structures in photonics.