The emerging picture of Earth's deep interior from seismic tomography indicates more complexity than previously thought. The presence of lateral anisotropy and heterogeneity in Earth's mantle highlights the need for fully anisotropic elasticity data from mineral physics. A breakthrough in high-frequency (gigahertz) ultrasound has resulted in transmission of pure-mode elastic shear waves into a high-pressure diamond-anvil cell using a P-to-S elastic-wave conversion. The full elastic tensor (c(ij)) of high-pressure minerals or metals can be measured at extreme conditions without optical constraints. Here we report the effects of pressure and composition on shear-wave velocities in the major lower-mantle oxide, magnesiowüstite-(Mg,Fe)O. Magnesiowüstite containing more than approximately 50% iron exhibits pressure-induced c(44) shear-mode softening, indicating an instability in the rocksalt structure. The oxide closer to expected lower-mantle compositions ( approximately 20% iron) shows increasing shear velocities more similar to MgO, indicating that it also should have a wide pressure-stability field. A complete sign reversal in the c(44) pressure derivative points to a change in the topology of the (Mg,Fe)O phase diagram at approximately 50-60% iron. The relative stability of Mg-rich (Mg,Fe)O and the strong compositional dependence of shear-wave velocities (and partial differential c(44)/ partial differential P) in (Mg,Fe)O implies that seismic heterogeneity in Earth's lower mantle may result from compositional variations rather than phase changes in (Mg,Fe)O.