  We introduce a categorical formalism for rewriting surface-embedded graphs.
Such graphs can represent string diagrams in a non-symmetric setting where we
guarantee that the wires do not intersect each other. The main technical
novelty is a new formulation of double pushout rewriting on graphs which
explicitly records the boundary of the rewrite. Using this boundary structure
we can augment these graphs with a rotation system, allowing the surface
topology to be incorporated.
