The magnetic and electronic structures of Fe4O5 have been investigated at ambient and high pressures via a combination of representation analysis, density functional theory (DFT+U) calculations, and M & ouml;ssbauer spectroscopy. A few spin configurations corresponding to the different irreducible representations have been considered. The total-energy calculations reveal that the magnetic ground state of Fe4O5 corresponds to an orthogonal spin order. Depending on the magnetic propagation vector k, two spin-ordered phases with minimal energy differences are realized. The lowest energy magnetic phase is related to k = (0, 0, 0) and is characterized by ferromagnetic ordering of iron magnetic moments at prismatic sites along the b-axis and antiferromagnetic ordering of iron moments at octahedral sites along the c-axis. For the k = (1/2, 0, 0) phase, the moments in the prisms are antiferromagnetically ordered along the b-axis and the moments in the octahedra are still antiferromagnetically ordered along the c-axis. Under high pressure, Fe4O5 exhibits magnetic transitions with the corresponding electronic transitions of the metal-insulator type. At a critical pressure P-C similar to 60 GPa, the Fe ions at the octahedral sites undergo a high-spin to low-spin state crossover with a decrease in the unit-cell volume of similar to 4%, while the Fe ions at the prismatic sites remain in the high-spin state up to 130 GPa. This site-dependent magnetic collapse is experimentally observed in the transformation of M & ouml;ssbauer spectra measured at room temperature and high pressures.