We develop a systematic framework to study the dispersion surfaces of Schrödinger operators $H=−\Delta+V$, where the potential $V \in C^\infty(\mathbb{R}^n,\mathbb{R})$ is both periodic with respect to a lattice $\Lambda \subset \mathbb{R}^n$ and respects its symmetries, and then use this framework to find Weyl points - long sought features in the band spectra of periodic operators. The framework consists of a series of results that can be used to analyze the operator $H$ where the lattice $\Lambda$ is arbitrary, which we prove using techniques from Floquet-Bloch theory and representation theory. As an application of this framework, we describe the generic structure of some singularities in the band spectrum of Schrödinger operators invariant under various three-dimensional cubic lattices, in the process proving a conjecture of [Haimo Guo, Meirong Zhang, and Yi Zhu. “Three-Fold Weyl Points for the Schrödinger Operator with Periodic Potentials”]. Lastly, using the band structure of Schrödinger operators invariant under the body-centered cubic lattice, we then introduce an equivariant magnetic field to produce the desired Weyl points.