We constructed global phase velocity maps including azimuthal anisotropy. Azimuthal anisotropy is expanded on a basis of generalized spherical harmonics, which makes the calculation of path integrals of the phase velocity particularly simple. It is generally accepted that the major difficulty in such modelling is determining the strength of the anisotropy relative to the isotropy. We propose a technique which finds the optimum relative weighting of the anisotropic terms prior to inversion. It is clear from our analyses that phase data require azimuthal anisotropy. We further find that Love wave data do not require a 2ψ term, whereas Rayleigh wave data need 2ψ and 4ψ terms. The main effect of azimuthal anisotropy upon the isotropic maps is a loss of power in the highest spherical harmonic degrees, resulting in an overall lower lateral resolution compared with a purely isotropic inversion for the same number of recovered parameters. The correlation of 2ψ and 4ψ components at different periods is relatively high, indicating a shallow source for the azimuthal anisotropy. Overall, fast 2ψ Rayleigh directions agree well with absolute plate motions in the hotspot reference frame.