  We study the behaviour of the spin-ellipticity radial tracks for 507 galaxies
from the Sydney AAO Multi-object Integral Field (SAMI) Galaxy Survey with
stellar kinematics out to $\geq1.5R_\text{e}$. 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 2-sigma 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 emphasises a clear
advantage to representing galaxies as a track, rather than a single point, in
spin-ellipticity parameter space.
