We describe quasi-phase-matched nonlinear interactions in materials that possess (4) over bar symmetry, such as GaAs. (4) over bar symmetry together with curved propagation geometries produce quasi phase matching ( QPM) without need of external domain inversions. To improve efficiency of nonlinear optical mixing, (4) over bar -QPM can be combined with resonant microcavities such as whispering-gallery-mode microdisks. All interacting waves must be resonant with the cavity, which leads to stringent tuning requirements. We show tuning behavior for second-harmonic generation in a GaAs microdisk cavity and estimate that, with milliwatt-level external pumping, approximately 0.1% power-conversion efficiency can be obtained with negligible higher-order nonlinear effects.