The discovery of a universal behavior in rare-earth (RE)-substituted perovskite BiFeO3 is reported. The structural transition from the ferroelectric rhombohedral phase to an orthorhombic phase exhibiting a double-polarization hysteresis loop and substantially enhanced electromechanical properties is found to occur independent of the RE dopant species. The structural transition can be universally achieved by controlling the average ionic radius of the A-site cation. Using calculations based on first principles, the energy landscape of BiFeO3 is explored, and it is proposed that the origin of the double hysteresis loop and the concomitant enhancement in the piezoelectric coefficient is an electric-field-induced transformation from a paraelectric orthorhombic phase to the polar rhombohedral phase.