Maps of orientation preference and selectivity, inferred from differential images of orientation (Blasdel, 1992), reveal linear organizations in patches, 0.5-1.0 mm across, where orientation selectivities are high, and where preferred orientations rotate linearly along one axis while remaining constant along the other. Most of these linear zones lie between the centers of adjacent ocular dominance columns, with their short iso-orientation slabs oriented perpendicular, in regions enjoying the greatest binocular overlap. These two-dimensional linear zones are segregated by one- and zero- dimensional discontinuities that are particularly abundant in the centers of ocular dominance columns, and that are also correlated with cytochrome oxidase-rich zones within them. Discontinuities smaller than 90 degrees extend in one dimension, as fractures, while discontinuities greater than 90 degrees are confined to points, in the form of singularities, that are generated when orientation preferences rotate continuously through +/- 180 degrees along circular paths. The continuous rotations through 180 degrees imply that direction preferences are not organized laterally in striate cortex. And they also ensure that preferences for all orientations converge at each singularity, with perpendicular orientations represented uniquely close together on opposite sides. The periodic interspersing of linear zones and singularities suggests that orientation preferences are organized by at least two competing schemes. They are optimized for linearity, along with selectivity and binocularity, in the linear zones, and they are optimized for density near singularities. Since upper-layer neurons are likely to have similarly sized dendritic fields in all regions (Lund and Yoshioka, 1991), those in the linear zones should receive precise information about narrowly constrained orientations, while those near singularities should receive coarse information about all orientations--very different inputs that suggest different perceptual functions.