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?

###update: 
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