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Misalignments between the rotation axis of stars and gas are an indication of external processes shaping galaxies throughout their evolution. Using observations of 3068 galaxies from the SAMI Galaxy Survey, we compute global kinematic position angles for 1445 objects with reliable kinematics and identify 169 (12 per cent) galaxies which show stellar-gas misalignments. Kinematically decoupled features are more prevalent in early-type/passive galaxies compared to late-type/star-forming systems. Star formation is the main source of gas ionization in only 22 per cent of misaligned galaxies; 17 per cent are Seyfert objects, while 61 per cent show Low-Ionization Nuclear Emission-line Region features. We identify the most probable physical cause of the kinematic decoupling and find that, while accretion-driven cases are dominant, for up to 8 per cent of our sample, the misalignment may be tracing outflowing gas. When considering only misalignments driven by accretion, the acquired gas is feeding active star formation in only ∼1/4 of cases. As a population, misaligned galaxies have higher Sérsic indices and lower stellar spin and specific star formation rates than appropriately matched samples of aligned systems. These results suggest that both morphology and star formation/gas content are significantly correlated with the prevalence and timescales of misalignments. Specifically, torques on misaligned gas discs are smaller for more centrally concentrated galaxies, while the newly accreted gas feels lower viscous drag forces in more gas-poor objects. Marginal evidence of star formation not being correlated with misalignment likelihood for late-type galaxies suggests that such morphologies in the nearby Universe might be the result of preferentially aligned accretion at higher redshifts. [ABSTRACT FROM AUTHOR]