Snap-through buckling instability of elastic shells can provide a variety of biological and artificial mechanical systems with an efficient strategy to generate rapid and powerful actuation. However, snapping spherical shells studied to date have typically been shallow and thus are dominantly prone to axisymmetric inversions. Here, we study diffusion-swelling stimulated snap-through inversion of bilayer shells of a wide range of depth to cover non-axisymmetric as well as axisymmetric modes. We first establish an analytical model of strain energy stored in axisymmetrically swelling shells, in order to predict the snap-through conditions based on energy minimization. Confirming that the strain energy can indicate the critical conditions for snap-through, we compare the conditions of axisymmetric and non-axisymmetric snap-through inversion using both experiments and numerical simulations. We find that differentially swelling bilayer shells snap-through with a time-lagged but increased energy release during inversion when buckled non-axisymmetrically rather than axisymmetrically.