We aim at a detailed description of the kinematic properties of the old (several Gyrs) late-type CO absorption star population among the Galactic Center (GC) cl uster stars. This cluster is composed of a central supermassive black hole (Sgr A*) and a self-gravitating system of stars. Understanding its kinematics thus of fers the opportunity to understand the dynamical interaction between a central p oint mass and the surrounding stars in general, especially in view of understand ing other galactic nuclei. We apply AO assisted near-infrared imaging and integral field spectroscopy using the instruments NAOS/CONICA and SINFONI at the VLT. We obtain proper motions fo r 5445 stars, 3D velocities for 664 stars, and acceleration limits (in the sky p lane) for 750 stars. Global kinematic properties are analyzed using velocity and velocity dispersion distributions, phase-space maps, two-point correlation func tions, and the Jeans equation. We detect for the first time significant cluster rotation in the sense of the ge neral 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_sta t+/-0.13_sys kpc. The distribution of 3D stellar speeds can be app roximated by local Maxwellian distributions. Kinematic modelling provides de projected 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 cluste r. Using upper limits on accelerations, we constrain the minimum line-of-sight d istances 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 su bstantial star cluster. Overall, the GC late-type cluster is well described as a uniform, isotropic, rotating, dynamically relaxed, phase-mixed system.