Shear has been known to exist for many years in the topographic structure of the primary visual cortex, but has received little attention in the modeling literature. Although the topographic map of V1 is largely conformal (i.e. zero shear), several groups have observed topographic shear in the region of the V1/V2 border. Furthermore, shear has also been revealed by anisotropy of cortical magnification factor within a single ocular dominance column. In the present paper, we make a functional hypothesis: the major axis of the topographic shear tensor provides cortical neurons with a preferred direction of orientation tuning. We demonstrate that isotropic neuronal summation of a sheared topographic map, in the presence of additional random shear, can provide the major features of cortical functional architecture with the ocular dominance column system acting as the principal source of the shear tensor. The major principal axis of the shear tensor determines the direction and its eigenvalues the relative strength of cortical orientation preference. This hypothesis is then shown to be qualitatively consistent with a variety of experimental results on cat and monkey orientation column properties obtained from optical recording and from other anatomical and physiological techniques. In addition, we show that a recent result of Das and Gilbert (Das, A., & Gilbert, C. D., 1997. Distortions of visuotopic map match orientation singularities in primary visual cortex. Nature, 387, 594-598) is consistent with an infinite set of parameterized solutions for the cortical map. We exploit this freedom to choose a particular instance of the Das-Gilbert solution set which is consistent with the full range of local spatial structure in V1. These results suggest that further relationships between ocular dominance columns, orientation columns, and local topography may be revealed by experimental testing.