ABSTRACT We study the behaviour of the spin-ellipticity radial tracks for 507 galaxies from the Sydney AAO Multiobject Integral Field (SAMI) Galaxy Survey with stellar kinematics out to ≥1.5Re. We advocate for a morpho-dynamical classification of galaxies, relying on spatially resolved photometric and kinematic data. We find the use of spin-ellipticity radial tracks is valuable in identifying substructures within a galaxy, including embedded and counter-rotating discs, that are easily missed in unilateral studies of the photometry alone. Conversely, bars are rarely apparent in the stellar kinematics but are readily identified on images. Consequently, we distinguish the spin-ellipticity radial tracks of seven morpho-dynamical types: elliptical, lenticular, early spiral, late spiral, barred spiral, embedded disc, and 2σ galaxies. The importance of probing beyond the inner radii of galaxies is highlighted by the characteristics of galactic features in the spin-ellipticity radial tracks present at larger radii. The density of information presented through spin-ellipticity radial tracks emphasizes a clear advantage to representing galaxies as a track, rather than a single point, in spin-ellipticity parameter space.