We describe a three-dimensional geometry in which both attractive and repulsive Casimir forces arise using ordinary metallic materials, as computed via an exact numerical method (no uncontrolled approximations). The geometry consists of a zipperlike, glide-symmetric structure formed by interleaved metal brakets attached to parallel plates–-because of the interleaving pattern, a net repulsive force can arise from acombination of attractive interactions. Depending on the separation, the perpendicular force between the plates and brackets varies from attractive (large separations) to repulsive (intermediate distances) and back to attractive (close separations), with one point of stable equilibrium in the perpendiculr direction. This geometry was motivated by a simple intuition of attractive interactions between surfaces, and so we also consider how a rough proximity-force approximation of pairwise attractions compares to the exact calculation.