Two closely related models of oriented self-avoiding walks (OSAWs) on a square lattice are studied. We use the pruned-enriched Rosenbluth method to determine numerically the phase diagram. Both models have three phases:
a tight-spiral phase in which the binding of parallel steps dominates, a collapsed phase when the binding of antiparallel steps dominates, and a free (open coil) phase. We show that the system features a first-order phase transition from the free phase to the tight-spiral phase, while both other transitions are continuous. The location of the phases is determined accurately. We also study turning numbers and gamma exponents in various regions of the phase diagram.