Deciding satisfiability and non-validity
For (propositional) intermediate logics, a decision procedure for satisfiability cannot generally be turned into a decision procedure for (non)validity, because all intermediate logics happen to have the same set of satisfiable formulas, namely, the set of classically satisfiable formulas, whereas some (a majority of) intermediate logics are undecidable.

(To make sense of the juxtaposition, all of the intermediate logics should be considered in the language of the weakest one, i.e., $\land$, $\lor$, $\to$, and $\neg$ or $\bot$.)

The coincidence of SAT's for these logics is a consequence of Glivenko theorem (which can be paraphrased by asserting that $\neg\varphi$ is provable classically iff it is provable intuitionistically, for any formula $\varphi$). See also this post.