The topologies permitted in joint ocular dominance (OD), orientation preference (OP), and direction preference (DP) maps in the primary visual cortex (V1) are considered, with the aim of finding a maximally symmetric periodic case that can serve as the basis for perturbations toward natural realizations. It is shown that mutual consistency of the maps selects just two possible such lattice structures, and that one of these is much closer to experiment than the other. This comprises a hexagonal lattice of alternating positive and negative OP singularities, with each unit cell or hypercolumn containing four such singularities, each radiating three DP discontinuities that follow OP contours and end at OP singularities of opposite sign. Other DP discontinuities emanate at 90 degrees to the midpoints of the ones that link OP singularities, and cross OP contours perpendicularly. These features explain experimentally observed relationships between DP discontinuities and OP contours, including sudden approximately 90-degree changes of direction in the former.