Seismological observations (SKS-wave polarizations, systematic P-residual variations, azimuthal dependence of P(n)- and surface-wave velocities or a dispersion of surface waves) are not consistent with isotropic, if laterally heterogeneous, upper-mantle structure. Therefore, an anisotropy should be considered as an a priori aspect of future large-scale studies of mantle structure. Most studies of anisotropy, however, have assumed horizontal or vertical axes of symmetry, but such orientations cannot explain bipolar patterns of spatial variations of P residuals, which we have observed at many seismological stations. On the basis of the petrophysical properties of real upper-mantle rocks we consider anisotropy formed either by hexagonal or by orthorhombic aggregates composed of olivine, orthopyroxene, and clinopyroxene. Rotations of the aggregates about vertical and horizontal axes allow us to find the three-dimensional orientations of symmetry axes that fit combinations of both P and S seismological observations in Central Europe an in western North America. The orientations with plunging symmetry axes (velocity extremes) seem to be consistent across large, spatially uniform tectonic units and change abruptly at important suture zones.