  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. 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. 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 SgrA* 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.
