Nonlinear dynamics of mutually injected twin-microring semiconductor lasers are studied numerically and experimentally. For the twin-microring laser, a bridge waveguide directly connecting two identical microrings is utilized to realize mutual optical injection or mode coupling. The coupling coefficient is related to the waveguide parameter and influences the mutual injection process dramatically. By adjusting the frequency offset between two microrings, four-wave mixing, injection locking, periodic, and multi-period oscillation states are observed from the experimental lasing spectra. The plentiful nonlinear dynamical states are induced by the mode interaction between the two microrings and can be well explained by the rate equations with mutual optical injection.