# Sufficient condition for a complexity class's closure under NP-reductions?

Let us say that there exists a $$\mathsf{NP}$$-reduction from a problem $$A$$ to another problem $$B$$ when there exists a non-deterministic, polynomial-time Turing machine $$T$$ such that for each $$a \in A$$, one of the possible outputs of $$T(a)$$ is in $$B$$ (and conversely, if one output $$T(a) \in B$$, then $$a \in A$$). $$\mathsf{NP}$$ reductions are discussed with relevant citations in the opening pages of this article.

A complexity class $$\mathcal{C}$$ is closed under $$\mathsf{NP}$$-reductions, if whenever a problem $$A$$ can be $$\mathsf{NP}$$-reduced to some $$B \in \mathcal{C}$$, we have $$A \in \mathcal{C}$$. (I think) equivalently, the class $$\mathcal{C}$$ is closed under $$\mathsf{NP}$$-reductions if whenever $$B$$ is $$\mathcal{C}$$-complete and there exists a $$\mathsf{NP}$$ reduction from $$A$$ to $$B$$, there also exists a polynomial-time (deterministic) many-one reduction from $$A$$ to $$B$$.

Of course, any class $$\mathcal{C}$$ that is closed under $$\mathsf{NP}$$-reductions must include $$\mathsf{NP}$$, but it isn't obvious to me that any class containing $$\mathsf{NP}$$ is closed under $$\mathsf{NP}$$-reductions. Indeed, on p. 545 of this article it is claimed that closure under $$\mathsf{NP}$$-reductions is nontrivial and that every class in the series $$\mathsf{NP} \subseteq \exists\mathbf{R} \subseteq \mathsf{PSPACE}$$ is closed under $$\mathsf{NP}$$-reductions.

Here is my question: What is a non-trivial condition on a complexity class $$\mathcal{C}$$, such that when $$\mathcal{C}$$ satisfies that condition, it is closed under $$\mathsf{NP}$$ reductions?