We present ab initio calculations of hidden magnetoelectric multipolar order in Cr₂O₃ and its iron-based analogue, α-Fe₂O₃. We show the presence of hidden antiferroically-ordered magnetoelectric multipoles in both the prototypical magnetoelectric material Cr₂O₃, and centrosymmetric α-Fe₂O₃, which has the same crystal structure as Cr₂O₃, but a different magnetic dipolar ordering. In turn, we predict anti-magnetoelectric effects, in which local magnetic dipole moments are induced in opposite directions under the application of an external electric field, to create an additional antiferromagnetic ordering. We confirm the predicted induced moments using first-principles calculations. Our results demonstrate the existence of hidden magnetoelectric multipoles leading to local linear magnetoelectric responses even in centrosymmetric materials, where a net bulk linear magnetoelectric effect is forbidden by symmetry. This archive contains input files and some output files necessary for creating all the data discussed in our paper with the title: Hidden orders and (anti-)Magnetoelectric Effects in Cr₂O₃ and α-Fe₂O₃.