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Finite-element micromagnetic simulations are used to study the chiral symmetry breaking of magnetic vortices in thin-film magnetic structures, caused by surface roughness. An asymmetry between vortices with different core polarizations has been experimentally observed for square-shaped platelets. E.g., the threshold fields for vortex core switching were found to differ for core up and down. This asymmetry was however not expected for these symmetrically-shaped structures, where both core polarizations should behave symmetrically. Three-dimensional finite element simulations are employed to show that a small surface roughness can break the symmetry between vortex cores pointing up and down. A relatively small sample roughness is found sufficient to reproduce the experimentally observed asymmetries. It arises from the lack of mirror-symmetry of the rough thin-film structures, which causes vortices with different chiralities to exhibit asymmetric dynamics.