From a sample of more than 100 remnants from major and minor hydrodynamic binary galaxy merger simulations (Cox 2004; Cox et al. 2005), we find that stellar remnants are mostly oblate while dark matter halos are mostly prolate or triaxial. Shapes are determined by iteratively diagonalizing a moment-of-inertia tensor. The preferred axes of the two shapes are almost always nearly perpendicular. This can be understood by considering the influence of angular momentum and dissipation during the merger. If binary major mergers of spiral galaxies are responsible for the formation of elliptical galaxies or some subpopulation of elliptical galaxies, then the galaxies can be be expected to be oblate and the dark matter halos prolate with the two preferred axes perpendicular to each other.