Binocular vision gives rise to a perceptual space that is endowed with a rich geometrical structure, which according to Luneburg is a non-Euclidean Riemannian geometry of constant Gaussian curvature. This hypothesis, together with certain psychophysical assumptions, provides a qualitative explanation of classical empirical phenomena. The Luneburg theory has been less successful in quantitatively predicting experimental data, a fact that has sometimes been taken as evidence against the presumed geometrical structure. Some of these shortcomings, however, may also be attributed to psychophysical assumptions that are independent of the geometry of visual space, and that were found to be violated systematically. We propose a generalization of Luneburg's psychophysical mapping that can account for these findings. The foundation of a psychologically significant recoordinatization of physical space is formulated within a conjoint measurement approach. It is shown that psychophysical invariances lead to restrictions on the possible form of the coordinate transformation, but its complete specification is left as an open problem. The proposed approach can at least partially explain some empirical observations that are considered to contradict Luneburg's conception of a psychophysical theory of binocular space perception.