Measurements of angular position and velocity vectors of the eye in three human and three monkey subjects showed that: (1) position vectors lie roughly in a single plane, in accordance with Listing's law, between and during saccades; (2) primary position of the eye is often far from the centre of the oculomotor range. (3) saccades have nearly-fixed rotation axes, which tilt out of Listing's plane in a systematic way depending on current eye position. Findings 1 and 3 show that saccadic control signals accurately reflect the properties of three-dimensional rotations, as predicted by a new quaternion model of the saccadic system; models that approximate rotational kinematics using vectorial addition and integration do not predict these findings.