  Situated in the so-called "island of inversion," the nucleus $^{32}$Mg is
considered as an archetypal example of the disappearance of magicity at $N=20$.
We report on high statistics in-beam spectroscopy of $^{32}$Mg 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 $^{32}$Mg were populated by knockout reactions starting
from $^{33}$Mg and $^{34}$Si, 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 $^{32}$Mg 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.
