A symmetric dual is one where there is equivalence: the dual of the dual is the primal.

But it is possible to have a "weak" duality that is assymmetric, where the dual encompasses the primal -- where there are regions of equivalence instead of a 1-to-1 equivalence.

I'm not sure if stream-duality fits this conceptual mold exactly but thank you for providing a counterpoint to my assertion. I'm going to think on it.