The variation of the kinematical properties of the Galactic thick disk with Galactic height Z is studied by means of 412 red giants observed in the direction of the south Galactic pole up to 4.5 kpc from the plane. We confirm the non-null mean radial motion toward the Galactic anticenter found by other authors, but we find that it changes sign at |Z| = 3 kpc, and the proposed inward motion of the local standard of rest alone cannot explain these observations. The rotational velocity decreases with |Z| by -30 km s(-1) kpc(-1), but the data are better represented by a power law with index 1.25, similar to that proposed from the analysis of Sloan Digital Sky Survey data. All the velocity dispersions increase with |Z|, but the vertical gradients are small. The dispersions grow proportionally, with no significant variation of the anisotropy. The ratio sigma(U)/sigma(W) = 2 suggests that the thick disk could have formed from a low-latitude merging event. The vertex deviation increases with Galactic height, reaching similar to 20 degrees at |Z| = 3.5 kpc. The tilt angle also increases, and the orientation of the ellipsoid in the radial-vertical plane is constantly intermediate between the alignment with the cylindrical and the spherical coordinate systems. The tilt angle at |Z| = 2 kpc coincides with the expectations of MOdified Newtonian Dynamics, but an extension of the calculations to higher |Z| is required to perform a conclusive test. Finally, between 2.5 and 3.5 kpc we detect deviations from the linear trend of many kinematical quantities, suggesting that some kinematical substructure could be present.