We report a new analysis of the stellar dynamics in the Galactic Centre, based on improved sky and line-of-sight velocities for more than 100 stars in the central few arcseconds from the black hole candidate SgrA*. The main results are as follows. (1) Overall, the stellar motions do not deviate strongly from isotropy. For those 32 stars with a determination of all three velocity components, the absolute, line-of-sight and sky velocities are in good agreement, consistent with a spherical star cluster. Likewise the sky-projected radial and tangential velocities of all 104 proper motion stars in our sample are also consistent with overall isotropy. (2) However, the sky-projected velocity components of the young, early-type stars in our sample indicate significant deviations from isotropy, with a strong radial dependence. Most of the bright Hei emission-line stars at separations from 1 to 10arcsec from SgrA* are on tangential orbits. This tangential anisotropy of the Hei stars and most of the brighter members of the IRS 16 complex is largely caused by a clockwise (on the sky) and counter-rotating (line of sight, compared to the Galaxy), coherent rotation pattern. The overall rotation of the young star cluster may be a remnant of the original angular momentum pattern in the interstellar cloud from which these stars were formed. (3) The fainter, fast-moving stars within ~1arcsec of SgrA* may be largely moving on radial or very elliptical orbits. We have so far not detected deviations from linear motion (i.e., acceleration) for any of them. Most of the SgrA* cluster members are also on clockwise orbits. Spectroscopy indicates that they are early-type stars. We propose that the SgrA* cluster stars are those members of the early-type cluster that happen to have small angular momentum, and thus can plunge to the immediate vicinity of SgrA*. (4) We derive an anisotropy-independent estimate of the Sun-Galactic Centre distance between 7.8 and 8.2kpc, with a formal statistical uncertainty of +/-0.9kpc. (5) We explicitly include velocity anisotropy in estimating the central mass distribution. We show how Leonard-Merritt and Bahcall-Tremaine mass estimates give systematic offsets in the inferred mass of the central object when applied to finite concentric rings for power-law clusters. Corrected Leonard-Merritt projected mass estimators and Jeans equation modelling confirm previous conclusions (from isotropic models) that a compact central mass concentration (central density >=10^12.6M_solarpc^-3) is present and dominates the potential between 0.01 and 1pc. Depending on the modelling method used, the derived central mass ranges between 2.6×10⁶ and 3.3×10⁶M_solar for R_solar=8.0kpc.