Dielectric metasurfaces supporting optical resonances have become a promising platform for quantum and nonlinear optics. However, resonant metasurfaces remain limited in their capacity to independently control the behavior of many distinct resonances despite efforts in computational optimization and inverse design. In this work, we overcome longstanding limitations by introducing a generalized rational design paradigm based on symmetry. Specifically, we use symmetry-broken metasurfaces with periodic "quadromer" lattices comprised of four nanostructures per unit cell to enable extensive control of multiple optical resonances. The rationally designed metasurfaces are readily fabricable, and we experimentally demonstrate metasurfaces that support up to four high Q-factor resonances with deliberately chosen free-space polarizations, spectral separations, and mode profiles. Our design paradigm may unlock new applications for multiresonant metasurfaces in quantum and nonlinear optics, optical sensing, and augmented reality displays.