The two-dimensional electron system at the interface between the insulating oxides LaAlO(3) and SrTiO(3) exhibits ferromagnetism, superconductivity and a range of unique magnetotransport properties. An open experimental challenge is to identify, out of the multitudinous energy bands predicted to exist at the interface, the key ingredients underlying its emergent transport phenomena. Here we show, using magnetotransport measurements, that a universal Lifshitz transition between d orbitals of different symmetries lies at the core of the observed phenomena. We find that LaAlO(3)/SrTiO(3) systems generically switch from one- to two-carrier transport at a universal carrier density, which is independent of the LaAlO(3) thickness and electron mobility. Interestingly, the maximum superconducting critical temperature occurs also at the Lifshitz density, indicating a possible connection between the two phenomena. A simple band model, allowing for spin-orbit coupling at the atomic level, connects the observed transition to a variety of previously reported properties. Our results demonstrate that the fascinating behaviour observed so far in these oxides follows from a small but fundamental set of bands.