Symmetry engineering is explicitly effective to manipulate and even create phases and orderings in strongly correlated materials. Flexural stress is universally practical to break the space-inversion or time-reversal symmetry. Here, by introducing strain gradient in a centrosymmetric antiferromagnet Sr_{2}IrO_{4}, the space-inversion symmetry is broken accompanying a nonequivalent O p-Ir d orbital hybridization along the z axis. Thus, an emergent polar phase and out-of-plane magnetic moment have been simultaneously observed in these asymmetric Sr_{2}IrO_{4} thin films, which both are absent in its ground state. Furthermore, upon the application of a magnetic field, such polarization can be controlled by modifying the occupied d orbitals through spin-orbit interaction, giving rise to a flexomagnetoelectric effect. This Letter provides a general strategy to artificially design multiple symmetries and ferroic orderings in strongly correlated systems.