  Background: The nucleus $^{32}$Mg ($N=20$ and $Z=12$) plays a central role in
the so-called "island of inversion" where in the ground states $sd$-shell
neutrons are promoted to the $fp$-shell orbitals across the shell gap,
resulting in the disappearance of the canonical neutron magic number $N=20$.
Purpose: The primary goals of this work are to extend the level scheme of
$^{32}$Mg, provide spin-parity assignments to excited states, and discuss the
microscopic structure of each state through comparisons with theoretical
calculations. Method: In-beam $\gamma$-ray spectroscopy of $^{32}$Mg was
performed using two direct-reaction probes, one-neutron (two-proton) knockout
reactions on $^{33}$Mg ($^{34}$Si). Final-state exclusive cross sections and
parallel momentum distributions were extracted from the experimental data and
compared with eikonal-based reaction model calculations combined with
shell-model overlap functions. Results: Owing to the remarkable selectivity of
the one-neutron and two-proton knockout reactions, a significantly updated
level scheme for $^{32}$Mg, which exhibits negative-parity intruder and
positive-parity normal states, was constructed. The experimental results were
confronted with four different nuclear structure models. Conclusions: In some
of these models, different aspects of $^{32}$Mg and the transition into the
island of inversion are well described. However, unexplained discrepancies
remain, and even with the help of these state-of-the-art theoretical
approaches, the structure of this key nucleus is not yet fully captured.
