This work is devoted to the analysis of the orbital ordering patterns and associated interatomic magnetic interactions in the centrosymmetric monoclinic structures of BiMnO₃, which have been recently determined experimentally. First, we set up an effective lattice fermion model for the manganese 3d bands and derive parameters of this model from first-principles electronic structure calculations. Then, we solve this model in terms of the mean-field Hartree–Fock method and obtain parameters of interatomic magnetic interactions between Mn ions. We argue that although the nearest-neighbor interactions favor the ferromagnetism, they compete with the longer range antiferromagnetic (AFM) interactions, the existence of which is directly related to the peculiar geometry of the orbital ordering pattern realized in BiMnO₃ below 474 K. These AFM interactions favor an AFM alignment, which breaks the inversion symmetry. The formation of the AFM phase is additionally assisted by the orbital degrees of freedom, which tend to adjust the nearest-neighbor magnetic interactions in the direction which further stabilizes this phase. We propose that the multiferroelectric behavior, which was observed in BiMnO₃, may be related to the emergence of the AFM phase under certain conditions.