Molecular dynamics simulations and first-principles electronic structure calculations are used to study the structural behavior of orthoenstatite, MgSiO3, at high pressures. The calculations suggest two possible high-pressure polymorphs of orthoenstatite, one with P21ca and the other with Pbca symmetry. Both polymorphs are structurally related to orthoenstatite. Molecular dynamics simulations reveal the displacive nature of the phase transitions between the three phases. Electronic structure calculations predict a phase transition from orthoenstatite to the metastable P21ca structure at 9 GPa, which may explain the anomalies in elastic and vibrational properties observed experimentally. A second metastable transition from the P21ca to the high-pressure Pbca structure may be observable above 20 GPa.