Strong interaction between the spin and orbital angular momentum of electrons in a material can give that material unusual quantum properties beyond what can be described by textbook band theory. It can lift the spin degeneracy of electronic bands and create a spin-momentum locking that can be a lever for electrical control of a material's magnetism. It creates a new state of matter: topological insulators, by twisting the energy ordering of electronic states. It adds complexity to how light interacts with the material, enabling some

means for pointing photoelectron spins in a chosen direction.

However, the way spin-orbit coupling is manifested in a material is shaped by the symmetries of the crystal. In presenting findings from topological insulators, this dissertation demonstrates how those symmetries can force a material to have multiple spin textures in momentum space, each coupled to a distinct atomic orbital. Further, it shows how changes to those symmetries can eliminate those orbital-dependent spin textures and leave a single overall spin texture. Lastly, it shows how symmetries can allow crystals to harbor hidden spin textures in certain parts of their real space unit cells even when the overall band structure is spin degenerate. This last point is demonstrated in a cuprate high temperature superconductor, in which the newly discovered spin structure could have profound implications for the nature of superconductivity.

The results presented in this dissertation come from spin- and angle-resolved photoemission measurements of several quantum materials using a uniquely efficient and high resolution spectrometer. The details of how the spectrometer works when coupled with a laboratory-based laser and how that enables new experiments that were otherwise infeasible are given.

This dissertation is organized into six chapters. Chapter 1 reviews atomic spin-orbit coupling and ways it can influence the properties of materials. Chapter 2 explains spin- and angle-resolved photoemission and the unique instrumentation used for experiments in this work. Chapter 3 details experiments that show how spin textures in materials can couple to different orbitals. Chapter 4 shows how that coupling can break down when symmetry no longer requires it. Chapter 5 presents surprising findings of a spin texture hidden in a cuprate superconductor. Chapter 6 presents discussion of future research directions on the study and manipulation of the spin-orbit interaction in materials.