We present a novel pair of numerical models of the interaction history between the Large and Small Magellanic Clouds (LMC and SMC, respectively) and our Milky Way (MW) in light of recent high-precision proper motions from the Hubble Space Telescope. Given the updated velocities, cosmological simulations of hierarchical structure formation favour a scenario where the Magellanic Clouds (MCs) are currently on their first infall towards our Galaxy. We illustrate here that the observed irregular morphology and internal kinematics of the Magellanic System (in gas and stars) are naturally explained by interactions between the LMC and SMC, rather than gravitational interactions with the MW. These conclusions provide further support that the MCs are completing their first infall to our system. In particular, we demonstrate that the Magellanic Stream, a band of H i gas trailing behind the Clouds 150° across the sky, can be accounted for by the action of LMC tides on the SMC before the system was accreted by the MW. We further demonstrate that the off-centre, warped stellar bar of the LMC, and its one-armed spiral can be naturally explained by a recent direct collision with its lower mass companion, the SMC. Such structures are key morphological characteristics of a class of galaxies referred to as Magellanic Irregulars, the majority of which are not associated with massive spiral galaxies. We infer that dwarf–dwarf galaxy interactions are important drivers for the morphological evolution of Magellanic Irregulars and can dramatically affect the efficiency of baryon removal from dwarf galaxies via the formation of extended tidal bridges and tails. Such interactions are not only important for the evolution of dwarf galaxies but also have direct consequences for the build-up of baryons in our own MW, as LMC-mass systems are believed to be the dominant building blocks of MW-type haloes.