  A classic exercise in the topology of surfaces is to show that, using handle
slides, every disc-band surface, or 1-vertex ribbon graph, can be put in a
canonical form consisting of the connected sum of orientable loops, and either
non-orientable loops or pairs of interlaced orientable loops. Motivated by the
principle that ribbon graph theory informs delta-matroid theory, we find the
delta-matroid analogue of this surface classification. We show that, using a
delta-matroid analogue of handle-slides, every binary delta-matroid in which
the empty set is feasible can be written in a canonical form consisting of the
direct sum of the delta-matroids of orientable loops, and either non-orientable
loops or pairs of interlaced orientable loops. Our delta-matroid results are
compatible with the surface results in the sense that they are their ribbon
graphic delta-matroidal analogues.
