Shape-dependent modes dominate the optical states of nanoparticles in the infrared spectra of ionic crystals and in the visible spectra of plasmonic metals. Here, the characteristic dispersion, or the shift in energy associated with the spatial period of surface charge oscillations, of surface modes governed by charges on the faces of prismatic nanoparticles is shown to follow the same form as the dispersion of the antisymmetric and cavity modes of infinite thin films and cylinders. Examination of the quasi-static eigenmodes of cubic particles with and without an internal octahedral cavity reveals the origins of this antisymmetric character and demonstrates self-coupling in the particles containing cavities. In these particles, interactions between the face and cavity modes give rise to distinctive optical states at energies approaching the longitudinal optical phonon frequency in ionic crystals or the bulk plasma frequency in metal nanoparticles.