Using first-principles calculations, we identify a magnetostructural effect in the BiFeO3-BiMnO3 nanocheckerboard that is not to be found in either the bulk parent compound or in BiFeO3-BiMnO3 superlattices with (001)-oriented Fe and Mn layers. The key role of the cation arrangement is explained by a simple model of the exchange coupling between cation spins, leading to magnetic frustration in the checkerboard. We also demonstrate that the atomic-scale checkerboard has a multiferroic ground state with the desired properties of each constituent material: polar and ferrimagnetic due to BiFeO3 and BiMnO3, respectively.