Idealized nanowire geometries assume stable sidewalls at right angles to the growth front. Here we report growth simulations that include a mix of nonorthogonal facet orientations, as for Au-catalyzed Si. We compare these with in situ microscopy observations, finding striking correspondences. In both experiments and simulations, there are distinct growth modes that accommodate the lack of right angles in different ways--one through sawtooth-textured sidewalls, the other through a growth front at an angle to the growth axis. Small changes in conditions can reversibly switch the growth between modes. The fundamental differences between these modes have important implications for control of nanowire growth.