Perovskite transition-metal oxides are networks of corner-sharing octahedra whose tilts and distortions are known to affect their electronic and magnetic properties. We report calculations on a model interfacial structure to avoid chemical influences and show that the symmetry mismatch imposes an interfacial layer with distortion modes that do not exist in either bulk material, creating new interface properties driven by symmetry alone. Depending on the resistance of the octahedra to deformation, the interface layer can be as small as one unit cell or extend deep into the thin film.