The broadband recording site at Warramunga (WRA) in the Northern Territory of Australia provides good coverage of seismic wave propagation through the upper mantle for sources in the earthquake belt through Indonesia and New Guinea. S waves recorded on the radial (SV) and tangential (SH) components are of comparable quality because the hard-rock recording site minimizes the influence of coupling to P on the radial component. Refracted S waves from the upper-mantle transition zone show a clear advance of SH wave arrivals compared with SV. Eleven polarization analyses of waves returned from the transition zone yield an average time shift of 2.3 s with the fast direction scattered about the transverse direction. Nine polarization measurements of waves returned from the top of the lower mantle yield an average time shift of 1.7 s, again with the fast direction near the transverse. No appreciable time differences are observed between the radial and transverse polarizations for paths refracted within the lithospheric lid. Because the observations of shear wave splitting in waves passing through the low-velocity zone, the transition zone, and the top of the lower mantle are not coherent in their absolute polarization, the cause cannot lie in azimuthal anisotropy at shallow depths under the WRA station. The most plausible explanation is transverse isotropy in shear within the low-velocity zone under the unusually thick mantle ''lid'' under Australia. A possible contribution may come from anisotropy in β-olivine at the top of the upper mantle transition zone. Transverse isotropy in the 200-km-thick layer below the lithosphere down to the transition zone with a 1% faster shear wave speeds for horizontal polarization compared with vertical polarization will explain the splitting data. For this asthenospheric region the level of anisotropy is quite reasonable and the polarization is consistent with lateral flow. The geometry of the available paths for waves propagating within the mantle lid is not sufficient to place constraints on the anisotropic properties of this heterogeneous and low-loss region. ¿ American Geophysical Union 1994