Nova LMC 2009a is confirmed as a Recurrent Nova (RN) from positional coincidence with nova LMC 1971b. The observational data set is one of the most comprehensive for any Galactic or extragalactic RN: optical and near-IR photometry from outburst until over 6 years later; optical spectra for the first 6 months, and Swift satellite Ultraviolet and X-ray observations from 9 days to almost 1 year post-outburst. We find $M_V = -8.4pm0.8_{mathrm{r}}pm0.7_{mathrm{s}}$ and expansion velocities between 1000 and 4000 km s$^{-1}$. Coronal line emission before day 9 indicates shocks in the ejecta. Strengthening of He II $lambda$4686 preceded the emergence of the Super-Soft Source (SSS) in X-rays at $sim63-70$ days, which was initially very variable. Periodic modulations, $P=1.2$ days, most probably orbital in nature, were evident in the UV and optical from day 43. Subsequently, the SSS shows an oscillation with the same period but with a delay of 0.28P. The progenitor system has been identified; the secondary is most likely a sub-giant feeding a luminous accretion disk. Properties of the SSS infer a white dwarf (WD) mass $1.1 mathrm{M}_odot lesssim M_{rm WD} lesssim 1.3 mathrm{M}_odot$. If the accretion occurs at constant rate, $dot{it{M}}_{rm acc} simeq 3.6^{+4.7}_{-2.5} times 10^{-7} mathrm{M}_odot$ yr$^{-1}$ is needed, consistent with nova models for an inter-eruption interval of 38 years, low outburst amplitude, progenitor position in the color-magnitude diagram, and spectral energy distribution at quiescence. We note striking similarities between LMC 2009a and the Galactic nova KT Eri, suggesting that KT Eri is a candidate RN.
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