  Delta-matroid theory is often thought of as a generalization of topological
graph theory. It is well-known that an orientable embedded graph is bipartite
if and only if its Petrie dual is orientable. In this paper, we first introduce
the concepts of Eulerian and bipartite delta-matroids and then extend the
result from embedded graphs to arbitrary binary delta-matroids. The dual of any
bipartite embedded graph is Eulerian. We also extend the result from embedded
graphs to the class of delta-matroids that arise as twists of binary matroids.
Several related results are also obtained.
