Maximum size of a source-sink reorientable set
If we replace u and v by larger stable sets in the example below, then we obtain an instance where the union of two sink-stable sets is larger than the maximum size of a source-sink reorientable set. Erika Bérczi-Kovács showed that the latter can be computed in polynomial time using minimum cost circulations.
By a result of Erdős, Frank, and Kun  (that can also be derived from Sebő ), the maximum size of the union of two sink-stable sets can be computed in polynomial time, so the validity of the characterization would imply a polynomial algorithm for determining the size (but not necessarily for finding a largest source-sink reorientable set).
The following simple example shows that the union of two sink-stable sets (u and v in the example) may not be source-sink reorientable.