[Abridged] We analysed two-dimensional maps of 48 early-type galaxies obtained with the SAURON and OASIS integral-field spectrographs using kinemetry, a generalisation of surface photometry to the higher order moments of the line-of-sight velocity distribution (LOSVD). In the SAURON sample, we find that 31% of early-type galaxies are single component systems. 91% of the multi-components systems have two kinematic subcomponents, the rest having three. In addition, 29% of galaxies have kinematically decoupled components, nuclear components with significant kinematic twists. We find that the velocity maps of fast rotators closely resemble those of inclined disks, except in the transition regions between kinematic subcomponents. In terms of E/S0 classification, this means that 74% of Es and 92% of S0s have components with disk-like kinematics. For the majority of fast rotators, the kinematic axial ratios are equal to or less than their photometric axial ratios, contrary to what is predicted with isotropic Jeans models viewed at different inclinations. The position angles of fast rotators are constant, while they vary abruptly in slow rotators. Velocity dispersion maps of face-on galaxies have shapes similar to the distribution of light. We constructed local (bin-by-bin) h3 - V/sigma and h4 - V/sigma diagrams from SAURON observations. We confirm the classical anti-correlation of h3 and Vsigma, but we also find that h3 is almost zero in some objects or even weakly correlated with V/sigma. The distribution of h4 for fast and slow rotators is mildly positive on average. The difference between slow and fast rotators is traceable throughout all moments of the LOSVD, with evidence for different intrinsic shapes and orbital contents and, hence, likely different evolutionary paths.
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