To investigate the role of non-commutative computations in the oculomotor system 3D eye movements were measured in 7 healthy subjects using a memory-contingent vestibulo-oculomotor paradigm. Subjects had to fixate a luminous point target that appeared briefly at an eccentricity of 20° in one of four diagonal directions in otherwise complete darkness. After a fixation period of about 1s, the subject was moved through a sequence of two rotations about mutually orthogonal axes in one of two orders (30°yaw followed by 30° pitch and vice versa in upright and 30° yaw followed by 20° roll and vice versa in both upright and supine orientations). We found that the change in ocular torsion induced by consecutive rotations about the yaw and the pitch axis depended on the order of rotations, as predicted by 3D rotation kinematics. Similarly, after rotations about the yaw and roll axis torsion depended on the order of rotations, but now due to the change in final head orientation relative to gravity. Quantitative analyses of these ocular responses revealed that the rotational VORs in far vision closely matched the predictions of 3D rotation kinematics. We conclude that the brain uses an optimal VOR strategy with the restriction of a reduced torsional position gain. This restriction implies a limited oculomotor range in torsion and systematic tilts of the angular eye velocity as a function of gaze direction.