  In this work, we analyze a sample of $\sim$4000 massive ($M_*\geq 10^{11}
M_\odot$ at $z=0$) galaxies in TNG300, the $(300 \mathrm{Mpc})^3$ box of the
IllustrisTNG simulation suite. We characterize the shape and kinematics of
these galaxies with a focus on the kinematic misalignment ($\Psi_\mathrm{int}$)
between the angular momentum (AM) and morphological major axis. We find that
the traditional purely shape- or kinematics-based classifications are
insufficient to characterize the diversity of our sample and define a new set
of classes based on the rates of change of the galaxies' morphological and
kinematic axes. We show that these classes are mostly stable over time and
correspond to six distinct populations of galaxies: the rapid AM reorienters
(58% of our sample), unsettled galaxies (10%), spinning disks (10%), twirling
cigars (16%), misaligned slow reorienters (3%), and regular prolate rotators
(galaxies that display major axis rotation; 2%). We demonstrate that the
most-recent significant (mass-ratio $\mu>1/10$) mergers of these galaxies are
the primary cause for their present-day properties and find that these mergers
are best characterized at the point of the satellite's final infall -- that is,
much closer to the final coalescence than has been previously thought. We show
that regular prolate rotators evolve from spinning disk progenitors that
experience a radial merger along their internal AM direction. Finally, we argue
that these regular prolate rotators are distinct from the similarly-sized
population of rapid AM reorienters with large $\Psi_\mathrm{int}$, implying
that a large $\Psi_\mathrm{int}$ is not a sufficient condition for major axis
rotation.
