Indices of diffusion anisotropy calculated from diffusion coefficients acquired in two or three perpendicular directions are rotationally variant. In living monkey brain, these indices severely underestimate the degree of diffusion anisotropy. New indices calculated from the entire diffusion tensor are rotationally invariant (RI). They show that anisotropy is highly variable in different white matter regions depending on the degree of coherence of fiber tract directions. In structures with a regular, parallel fiber arrangement, water diffusivity in the direction parallel to the fibers (D| ≈ 1400–1800 × 10−6 mm2/s) is almost 10 times higher than the average diffusivity in directions perpendicular to them ((D + D⊥′)/2 ≈ 150–300 × 10−6 mm2/s), and is almost three times higher than previously reported. In structures where the fiber pattern is less coherent (e.g., where fiber bundles merge), diffusion anisotropy is significantly reduced. However, RI anisotropy indices are still susceptible to noise contamination. Monte Carlo simulations show that these indices are statistically biased, particularly those requiring sorting of the eigenvalues of the diffusion tensor based on their magnitude. A new intervoxel anisotropy index is proposed that locally averages inner products between diffusion tensors in neighboring voxels. This “lattice” RI index has an acceptably low error variance and is less susceptible to bias than any other RI anisotropy index proposed to date.