We investigate the exact dynamics of a system of two independent harmonic oscillators coupled through their angular momentum. The exact analytic solution of the equations of motion for the field operators is derived, and the conditions for dynamical stability are obtained. As for the application, we examine the emergence of squeezing and mode entanglement for an arbitrary separable coherent initial state. It is shown that close to instability, the system develops considerable entanglement, which is accompanied with simultaneous squeezing in the coordinate of one oscillator and the momentum of the other oscillator. In contrast, for weak coupling away from instability, the generated entanglement is small, with weak alternating squeezing in the coordinate and momentum of each oscillator. Approximate expressions describing these regimes are also provided.