Aims. We aim at a detailed description of the kinematic properties of the old, (several Gyrs) late-type CO-absorption star population among the Galactic centre (GC) cluster stars. This cluster is composed of a central supermassive black hole (Sgr A*) and a self-gravitating system of stars. Understanding its kinematics thus offers the opportunity to understand the dynamical interaction between a central point mass and the surrounding stars in general, especially in view of understanding other galactic nuclei. Methods: We applied AO-assisted, near-infrared imaging and integral-field spectroscopy using the instruments NAOS/CONICA and SINFONI at the VLT. We obtained proper motions for 5445 stars, 3D velocities for 664 stars, and acceleration limits (in the sky plane) for 750 stars. Global kinematic properties were analysed using velocity and velocity dispersion distributions, phase-space maps, two-point correlation functions, and the Jeans equation. Results: We detect for the first time significant cluster rotation in the sense of the general Galactic rotation in proper motions. Out of the 3D velocity dispersion, we derive an improved statistical parallax for the GC of R0 = 8.07 ± 0.32_stat ± 0.13_sys kpc. The distribution of 3D stellar speeds can be approximated by local Maxwellian distributions. Kinematic modelling provides deprojected 3D kinematic parameters, including the mass profile of the cluster. We find an upper limit of 4% for the amplitude of fluctuations in the phase-space distribution of the cluster stars compared to a uniform, spherical model cluster. Using upper limits on accelerations, we constrain the minimum line-of-sight distances from the plane of Sgr A* of five stars located within the innermost few (projected) arcsec. The stars within 0.7'' radius from the star group IRS13E do not co-move with this group, making it unlikely that IRS13E is the core of a substantial star cluster. Overall, the GC late-type cluster is described well as a uniform, isotropic, rotating, dynamically relaxed, phase-mixed system. Based on observations at the Very Large Telescope (VLT) of the European Southern Observatory (ESO), Cerro Paranal, Chile.