Using a generalized anisotropic tokamak equilibrium and an exact guiding centre drift formulation, the effect of parallel and perpendicular anisotropy on the toroidal precession drift is investigated. Significant differences between parallel and perpendicular pressure anisotropy are observed. While the Shafranov shift is not sensitive to the ratio of the parallel and perpendicular pressures p⊥/p∥, the deepening of the magnetic well is found to be sensitive to p⊥/p∥. Here, the diamagnetic effect identified by Connor et al 1983 Nucl. Fusion 23 1702 is generalized and found to depend crucially on the deposition of the energetic ions on which the equilibrium depends, and leads to test particle precessional drifts that depend sensitively on pitch angle.