We have determined the radial velocity-dispersion profiles (VDPs) of 72 clusters, each with over 50 confirmed members. The velocity dispersion sigma(v) in clusters is found to be a strong function of radius; the largest gradients generally occur in the inner 0.5 h(-1) Mpc. In 27 clusters the VDP is significantly centrally peaked, while in 11 cases we find an 'inverted' profile, i.e. sigma(v) increases with distance from the centre in the inner 0.5 h(-1) Mpc. In four other clusters, whose VDPs were unclassifiable, we find hints of a central inversion of sigma(v). Several tests indicate that the observed fractions of peaked and inverted VDPs cannot be explained as the result of noise in the data alone. It is possible to reproduce most of the global features of the VDPs with semi-analytical spherical models. The centrally peaked profiles require density profiles with slopes as steep as-3.5 and core radii as small as 0.03 h(-1) Mpc, whereas the inverted profiles require core radii larger than 0.2 h(-1) Mpc. The cluster mass profiles derived from the data confirm this relation between profile shape and core radius. The most peaked VDPs are not associated with cooling flows, but they do show a significant correlation with the presence of a central giant galaxy. With these models we cannot explain the inverted profiles as the result of mass segregation alone, nor as the result of a variation in the M/L-ratio or a cusp in the central density. The inverted profiles seem to require unusually strong circularization of orbits (beta < -1.5), which is hard to reconcile with other calculations on the dynamics of clusters. An alternative explanation is that inverted profiles result from anisotropic projection effects of structures in the core. Because in 12 clusters the shape of the VDP is found to depend on the magnitude limit, we test all 53 clusters for which a sufficient number of magnitudes are available for signs of luminosity segregation (LS). In 12 clusters we find a signal of LS that is significant at the 99 per cent level, although the effect is generally limited to the five brightest galaxies. In 11 other clusters we find signs of an opposite effect. There is hardly any correlation between signs of LS and other dynamically relevant properties. In all clusters (either with or without signs of LS) the velocity dispersions are in agreement with the X-ray temperatures (i.e. beta = 1), for the entire galaxy population as well as for the early-type galaxies or the galaxies with projected distances from the centre less than 0.5 h(-1) Mpc. The median relaxation time (orbital-decay time) for the inner 0.5 h(-1) Mpc, where the differences between the various VDPs are most prominent, is on average 0.45t(H), and ranges from 0.3t(H) for the peaked profiles to 0.6t(H) for the inverted VDPs. The region where the relaxation time is 0.2t(H) is within the errors equal to 0.27 h(-1) Mpc for all clusters, regardless of the type of VDP. Hence, the differences between the profiles are unlikely to be due to large differences in the relaxation time.
