The (100) surface of magnetic gamma'-Fe4N films epitaxially grown on Cu(100) shows a (2 x 2)p4gm reconstruction. The atom positions determined experimentally are properly accounted for by first-principles total energy calculations, which further reveal that the (2 x 2)p4gm reconstruction requires the topmost Fe4N2 plane to be magnetic and effectively decoupled from the rest of the crystal and that it originates from the perpendicular crossing of one-dimensionally dimerized Fe chains.