We characterize a system consisting of a two-mode waveguide coupled to a single-mode microring resonator possibly presenting a nonlinear response of Kerr type. By using the scattering parameter formalism extended to the multimode domain, we show that in the linear regime and for an ideally transparent medium, each resonance of the system can be exploited to perform complete even-to-odd (respectively, odd-to-even) modal conversion. Moreover, when the Kerr nonlinearity is effective, the microring enables a power-dependent modal switching mediated by phase bistability. Thanks to its mode-processing capabilities, this configuration is suitable to find application as a functional building-block in mode-division multiplexing (MDM) photonic integrated circuits.