A common notion of clockwise can help in planar rendezvous

Two players are lost in a grid of city streets and wish to meet as soon as possible. Knowing only the distribution of the other's initial location (two nodes away in one of the four compass directions), how do they move from intersection to intersection (between nodes of the lattice Z²) to achieve this? We assume that they do not have common compass directions to coordinate on, but that they can use their common notion of clockwise. We show that the latter, realistic assumption, can aid them in expediting their meeting (relative to a previous rendezvous problem which did not allow this). We also solve the easier ‘streets and avenues’ version of the problem, in which the players can distinguish between the axes (between streets and avenues). We discover several new phenomena which have not been seen before in planar rendezvous.