We study how numerically predicted seismic anisotropy in the upper mantle is affected by several common assumptions about the rheology of the convecting mantle and deformation-induced lattice preferred orientations (LPO) of minerals. We also use these global circulation and texturing models to investigate what bias may be introduced by assumptions about the symmetry of the elastic tensor for anisotropic mineral assemblages. Maps of elasticity tensor statistics are computed to evaluate symmetry simplifications commonly employed in seismological and geodynamic models. We show that most of the anisotropy predicted by our convection-LPO models is captured by estimates based on a best fitting hexagonal symmetry tensor derived from the full elastic tensors for the computed olivine:enstatite LPOs. However, the commonly employed elliptical approximation does not hold in general. The orientations of the best fitting hexagonal symmetry axes are generally very close to those predicted for finite strain axes. Correlations between hexagonal anisotropy parameters for P and S waves show simple, bilinear relationships. Such relationships can reduce the number of free parameters for seismic inversions if this information is included a priori. The match between our model predictions and observed patterns of anisotropy supports earlier, more idealized studies that assumed laboratory-derived mineral physics theories and seismic measurements of anisotropy could be applied to study mantle dynamics. The match is evident both in agreement between predicted LPO at selected model sites and that measured in natural samples, and in the global pattern of fast seismic wave propagation directions.