Background: The nucleus Mg32 (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 Mg32, provide spin-parity assignments to excited states, and discuss the microscopic structure of each state through comparisons with theoretical calculations. Method: In-beam γ-ray spectroscopy of Mg32 was performed using two direct-reaction probes: one-neutron (two-proton) knockout reactions on Mg33 (Si34). 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 Mg32, 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 Mg32 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.