In this dissertation, I will summarize two very different experiments involving layered transition metal dichalcogenides, along with the technical information required to understand and reproduce them. The first experiment concerns the optical properties of a single molecular layer of MoSe2. Electrons in MoSe2 have a hidden internal state, called the "valley" degree of freedom, related to the existence of multiple degenerate low-energy states. In other semiconductors with valley degeneracy, e.g. silicon and germanium, the valley state of electrons is hard to probe, and most experiments reveal only valley-averaged quantities. However, the broken inversion symmetry of a MoSe2 monolayer allows for a curious property -- the valley state of electrons can be read-out and manipulated with circularly polarized light. In our experiments, we used this property to measure how the magnetic moment of the exciton (a bound electron-hole pair) depends on its valley state. We measured polarized-resolved luminescence spectra in a magnetic field, and looked at the energy difference between excitons in the different valleys. Our results show that magnetic field can be used to break valley degeneracy in monolayer MoSe2, and also give important input (the exciton magnetic moment) for band structure models. The second experiment concerns spin transport across, or perhaps at, the interface between the transition metal dichalcogenide WTe2 and a ferromagnet, permalloy. By flowing current in the plane of a WTe2/permalloy stack, we are able to inject spins into the permalloy and generate a torque on its magnetic moment. This phenomena -- known a spin-orbit torque -- has been well studied in other heavy-metal/ferromagnet bilayers. The new idea of our work is to use a low-symmetry crystal as the spin-generation layer. Depositing permalloy onto WTe2 breaks the combined rotation-translation symmetry of the WTe2 crystal, resulting in a remarkably low symmetry stack. This allows for new torques forbidden by symmetry in conventional devices, corresponding to injection of spins with their moment out of the sample plane. Out-of-plane spin injection is ideal for the manipulation of technologically-relevant magnets with perpendicular anisotropy, so our results open new possibilities for non-volatile magnetic memory technologies. Although these two experiments concern separate realms of physics -- optics and spin transport -- they are connected by the theme of broken crystal symmetries. In both cases, the interplay of reduced crystal symmetry and spin-orbit coupling leads to additional control over internal electron states. Another theme is the utility of mechanical exfoliation to prepare high-quality crystalline samples, especially of low-symmetry single-crystal films and containing refractory elements.