A fixed set of vertices in the plane may have multiple planar straight-line triangulations in which the degree of each vertex is the same. As such, the degree information does not completely determine the triangulation. We show that even if we know, for each vertex, the number of neighbors in each of the four cardinal directions, the triangulation is not completely determined. In fact, we show that counting such triangulations is #P-hard via a reduction from #3-regular bipartite planar vertex cover.