Photonic-crystal ring resonators, in which periodic refractive-index modulations couple counter-propagating modes, have become a widely used platform for nonlinear photonics, offering mode-by-mode dispersion engineering. When several modulation components are superimposed, their relative angular alignment (phase) is an additional degree of freedom whose impact on the nonlinear dynamics has remained unclear. Here we show that the relative phases between the modulation patterns have a decisive impact. Through the interference of the nonlinear processes in forward and backward direction, they add a phase and mode selection rule to the established phase-matching conditions, which can fully suppress nonlinear coupling. They also impose a relative spectral chirp between forward- and backward-propagating waveforms. We illustrate these effects for parametric oscillation and dissipative Kerr solitons. Our results extend the concept of phase matching to resonators with interfering nonlinear pathways and establish the alignment of index gratings as a design dimension for nonlinear microresonators.