We present kinematical analyses of 22 Galactic globular clusters using the Hubble Space Telescope proper motion (HSTPROMO) catalogues recently presented in Bellini et al. (2014). For most clusters, this is the first proper-motion study ever performed, and, for many, this is the most detailed kinematic study of any kind. We use cleaned samples of bright stars to determine binned velocity-dispersion and velocity-anisotropy radial profiles and two-dimensional velocity-dispersion spatial maps. Using these profiles, we search for correlations between cluster kinematics and structural properties. We find that: (1) more centrally-concentrated clusters have steeper radial velocity-dispersion profiles; (2) on average, at 1σ confidence in two dimensions, the photometric and kinematic centres of globular clusters agree to within ~1", with a cluster-to-cluster rms of 4" (including observational uncertainties); (3) on average, the cores of globular clusters have isotropic velocity distributions to within 1% (σ_t/σ_r = 0.992 +/- 0.005), with a cluster-to-cluster rms of 2% (including observational uncertainties); (4) clusters generally have mildly radially anisotropic velocity distributions (σ_t/σ_r ~ 0.8-1.0) near the half-mass radius, with bigger deviations from isotropy for clusters with longer relaxation times; (5) there is a relation between σ_minor/σ_major and ellipticity, such that the more flattened clusters in the sample tend to be more anisotropic, with σ_minor/σ_major ~ 0.9-1.0. Aside from these general results and correlations, the profiles and maps presented here can provide a basis for detailed dynamical modelling of individual globular clusters. Given the quality of the data, this is likely to provide new insights into a range of topics concerning globular cluster mass profiles, structure, and dynamics. ; Comment: 22 pages, 18 figures, 4 tables, ApJ accepted