Situated in the so-called “island of inversion,” the nucleus 32Mg is considered as an archetypal example of the disappearance of magicity at N=20. We report on high statistics in-beam spectroscopy of 32Mg with a unique approach, in that two direct reaction probes with different sensitivities to the underlying nuclear structure are employed at the same time. More specifically, states in 32Mg were populated by knockout reactions starting from 33Mg and 34Si, lying inside and outside the island of inversion, respectively. The momentum distributions of the reaction residues and the cross sections leading to the individual final states were confronted with eikonal-based reaction calculations, yielding a significantly updated level scheme for 32Mg and spin-parity assignments. By fully exploiting observables obtained in this measurement, a variety of structures coexisting in 32Mg was unraveled. Comparisons with theoretical predictions based on shell-model overlaps allowed for clear discrimination between different structural models, revealing that the complete theoretical description of this key nucleus is yet to be achieved.