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Neutron scattering has a proud history of elucidating certain kinds of structural phase transitions, but sliding layer transitions such as those in MoTe₂ have been relatively neglected. On cooling, the monoclinic 1T’-MoTe₂ transitions into the orthorhombic T_d-MoTe₂, which has received much attention since it was reported to be a Weyl semimetal and to exhibit extreme magnetoresistance. MoTe₂ structures can, to a good approximation, be built from sequences of two symmetry-equivalent stacking operations, with transitions occurring via layer sliding between different stackings. Thus, a wide variety of nearly-degenerate structures are conceivable, and our elastic neutron scattering studies show that changes in stacking with temperature in MoTe₂ are, indeed, complex. Both order-to-order and order-to-disorder transitions exist along the T_d-1T' thermal hysteresis loop. A pseudo-orthorhombic T_d* phase with a four-layer unit cell appears only on warming. T_d* is centrosymmetric, and the order-to-order transitions between T_d and T_d* may make a more convenient topological switch than the disordered transitions to and from 1T’. The kink in resistivity vs. temperature on warming is primarily due to the onset of T_d*, and the residual hysteresis in the resistivity toward the temperature extremes is likely related to changes in the presence of 1T’- or T_d-phase twin domain boundaries. Changes in stacking have a subtle effect on low-energy shear phonon modes, as seen from inelastic neutron scattering. A multitude of ways of influencing these transitions are known; we will discuss how both W-substitution and pressure drive the transition toward a simpler phase coexistence behavior, though with opposite effects on transition temperature or the 1T’ β angle. We will discuss changes in band structure with pressure and strain. Finally, we will discuss open questions concerning the cause of the transition.