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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 ($Ψ_\mathrm{int}$) 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 $μ>1/10$) 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 $Ψ_\mathrm{int}$, implying that a large $Ψ_\mathrm{int}$ is not a sufficient condition for major axis rotation.