Let D=(V,E) be a directed graph that is the union of two edge-disjoint directed spanning trees. Suppose that
There no subset X of vertices so that there is precisely one directed edge from X to its complement and one directed edge from the complement of X to X.
Is it true that D has a directed spanning tree T, T is a subset of E, such that E-T is also a directed spanning tree and reversing the orientation of each edge of T results in a strongly connected digraph?
The answer is NO as pointed out by an example by Maria Chudnovsky and Paul Seymour, who also pointed out additional necessary conditions. A remaining question is:
Problem: Find a characterization of directed graphs $D$ that has a spanning tree $T$, such that $E-T$ is also a spanning tree and reversing the orientation of each edge of $T$ results in a strongly connected digraph