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We explore the structural and kinematic properties of the outskirts of the Large Magellanic Cloud (LMC) using data from the Magellanic Edges Survey (MagES) and Gaia EDR3. Even at large galactocentric radii ($8^\circ<R<11^\circ$), we find the north-eastern LMC disk is relatively unperturbed: its kinematics are consistent with a disk of inclination ~$36.5^\circ$ and line-of-nodes position angle ~$145^\circ$ east of north. In contrast, fields at similar radii in the southern and western disk are significantly perturbed from equilibrium, with non-zero radial and vertical velocities, and distances significantly in front of the disk plane implied by our north-eastern fields. We compare our observations to simple dynamical models of the Magellanic/Milky Way system which describe the LMC as a collection of tracer particles within a rigid potential, and the Small Magellanic Cloud (SMC) as a rigid Hernquist potential. A possible SMC crossing of the LMC disk plane ~400 Myr ago, in combination with the LMC's infall to the Milky Way potential, can qualitatively explain many of the perturbations in the outer disk. Additionally, we find the claw-like and arm-like structures south of the LMC have similar metallicities to the outer LMC disk ([Fe/H]~-1), and are likely comprised of perturbed LMC disk material. The claw-like substructure is particularly disturbed, with out-of-plane velocities >60 km s$^{-1}$ and apparent counter-rotation relative to the LMC's disk motion. More detailed N-body models are necessary to elucidate the origin of these southern features, potentially requiring repeated interactions with the SMC prior to ~1 Gyr ago.