Superconductivity in cuprate superconductors is associated to a doped Mott insulator scenario and this antiferromagnetic insulating state is considered to be universal. The phase diagram of cuprate superconductors aims at the fundamental comparison of hole- and electron doping into CuO2 planes. While these CuO2 planes are, indeed, common among all cuprate superconductors, variations of the coordination of the Cu2+ ions are entirely ignored. It is well established that superconductivity is induced by hole doping to the CuO2 planes with 5- or 6-fold coordinated copper. The induction of superconductivity in cuprates with 4-fold coordinated copper is – against common understanding – not associated to doping but to elimination of defects. Cuprates with 4-fold coordinated copper are metals per se and this fact has not been recognized . It would therefore be revealing if one could alter the copper coordination by keeping chemical elements as well as their stoichiometry constant while solely varying the coordination of copper. That is what we did using La2CuO4. Molecular beam epitaxy allows to explore thermodynamic synthesis conditions entirely inaccessible by other synthesis methods. In particular, low synthesis temperatures are key. We synthesized La2CuO4 with three different copper coordination: 6-fold coordination (T-phase), 5-fold coordination (T*-phase), and 4-fold coordination (T’-phase). All three phases were grown coherently onto SrLaAlO4, DyScO3, and PrScO3 substrates, respectively. Among the three isomers of La2CuO4 the T-phase (6-fold coordinated copper) is the thermodynamically stable polymorph and its electronic response is well known. So, it is not surprising that T-La2CuO4 is insulating [Fig. 1(a)]. The T*-phase of La2CuO4 is, however, a new material as it is not a thermodynamic stable polymorph. So far we shed some light on its electronic response (it is insulating) and magnetic ordering phenomena are currently investigated by low-energy muon spin resonance spectroscopy. The T’-phase of La2CuO4 is also only accessible by low-temperature synthesis methods and its electronic response strongly deviates from its isomers. T’-La2CuO4 is a superconductor [2, 3].