The dependence of seismic wavespeeds on propagation or polarization direction, called seismic anisotropy, is a relatively direct indicator of mantle deformation and flow. Mantle seismic anisotropy is often inferred from measurements of shear-wave splitting. A number of standard techniques to measure shear-wave splitting have been applied globally; for example, *KS splitting is often used to measure upper mantle anisotropy. In order to obtain robust constraints on anisotropic geometry, it is necessary to sample seismic anisotropy from different directions, ideally using different seismic phases with different incidence angles. However, many standard analysis techniques can only be applied for certain epicentral distances and source-receiver ge-ometries. In this work, we apply a "wavefield differencing" approach to (systematically) understand what parts of the seismic wavefield are most affected by seismic anisotropy in the mantle. We systematically analyze differences between synthetic global wavefields calculated for isotropic and anisotropic input models, incorporating seismic anisotropy at different depths. Our results confirm that the seismic phases that are commonly used in splitting techniques are indeed strongly influenced by mantle anisotropy. However, we also identify less commonly used phases whose waveforms reflect the effects of anisotropy. For example, PS is strongly affected by upper mantle seismic anisotropy. We show that PS can be used to fill in gaps in global coverage in shear wave splitting datasets (for example, beneath ocean basins). We find that PcS is also a promising phase, and present a proof-of-concept example of PcS splitting analysis across the contiguous United States using an array processing approach. Because PcS is recorded at at much shorter distances than *KS phases, PcS splitting can therefore fill in gaps in backazimuthal coverage. The insights provided by a wavefield differencing approach provide promising new strategies for improving our ability to detect and characterize seismic anisotropy in the mantle.