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We analysed two‐dimensional maps of 48 early‐type galaxies obtained with the SAURON and OASIS integral‐field spectrographs using kinemetry, a generalization of surface photometry to the higher order moments of the line‐of‐sight velocity distribution (LOSVD). The maps analysed include: reconstructed image, mean velocity, velocity dispersion,  h  3  and  h  4  Gauss–Hermite moments. Kinemetry is a good method to recognize structures otherwise missed by using surface photometry, such as embedded discs and kinematic subcomponents. In the SAURON sample, we find that 31 per cent of early‐type galaxies are single component systems. 91 per cent of the multicomponents systems have two kinematic subcomponents, the rest having three. In addition, 29 per cent of galaxies have kinematically decoupled components, nuclear components with significant kinematic twists. We differentiate between slow and fast rotators using velocity maps only and find that fast‐rotating galaxies contain discs with a large range in mass fractions to the main body. Specifically, we find that the velocity maps of fast rotators closely resemble those of inclined discs, except in the transition regions between kinematic subcomponents. This deviation is measured with the kinemetric   k  5 / k  1   ratio, which is large and noisy in slow rotators and about 2 per cent in fast rotators. In terms of E/S0 classification, this means that 74 per cent of Es and 92 per cent of S0s have components with disc‐like kinematics. We suggest that differences in   k  5 / k  1   values for the fast and slow rotators arise from their different intrinsic structure which is reflected on the velocity maps. 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. Velocity dispersion maps of the edge‐on fast rotators and all slow rotators show differences which can only be partially explained with isotropic models and, in the case of fast rotators, often require additional cold components. We constructed local (bin‐by‐bin)   h  3 – V /σ  and   h  4 – V /σ  diagrams from SAURON observations. We confirm the classical anticorrelation of  h  3  and   V /σ , but we also find that  h  3  is almost zero in some objects or even weakly correlated with   V /σ . The distribution of  h  4  for fast and slow rotators is mildly positive on average. In general, fast rotators contain flattened components characterized by a disc‐like rotation. 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.

The SAURON project – XII. Kinematic substructures in early‐type galaxies: evidence for discs in fast rotators