ABSTRACT Recent cosmological hydrodynamical simulations suggest that integral field spectroscopy can connect the high-order stellar kinematic moments h3 (∼skewness) and h4 (∼kurtosis) in galaxies to their cosmological assembly history. Here, we assess these results by measuring the stellar kinematics on a sample of 315 galaxies, without a morphological selection, using two-dimensional integral field data from the SAMI Galaxy Survey. Proxies for the spin parameter ( ) and ellipticity ( ) are used to separate fast and slow rotators; there exists a good correspondence to regular and non-regular rotators, respectively, as also seen in earlier studies. We confirm that regular rotators show a strong h3 versus anti-correlation, whereas quasi-regular and non-regular rotators show a more vertical relation in h3 and . Motivated by recent cosmological simulations, we develop an alternative approach to kinematically classify galaxies from their individual h3 versus signatures. Within the SAMI Galaxy Survey, we identify five classes of high-order stellar kinematic signatures using Gaussian mixture models. Class 1 corresponds to slow rotators, whereas Classes 2-5 correspond to fast rotators. We find that galaxies with similar values can show distinctly different signatures. Class 5 objects are previously unidentified fast rotators that show a weak h3 versus anti-correlation. From simulations, these objects are predicted to be disk-less galaxies formed by gas-poor mergers. From morphological examination, however, there is evidence for large stellar disks. Instead, Class 5 objects are more likely disturbed galaxies, have counter-rotating bulges, or bars in edge-on galaxies. Finally, we interpret the strong anti-correlation in h3 versus as evidence for disks in most fast rotators, suggesting a dearth of gas-poor mergers among fast rotators.