Let $J(C)$ be the jacobian of a hyperelliptic curve $C$ of genus 2 defined over finite field $\mathbb{F}_q$. Let $\Theta$ be the image of the curve on the Jacobian under the embedding $P \mapsto P - \mathcal{O}$, which is also known as the theta divisor.

Do we know something about the structure of the following set:
$$ J(C)_{\Theta}[n] := \{D \in J(C) : nD \in \Theta \}$$
Does this set have a name? Like "*Theta n-torsion points*"?

Clearly, the n-torsion points defined as $$ J(C)[n] := \{D \in J(C) : nD = \mathcal O \}$$ is a subset of $J(C)_{\Theta}[n]$.

I could not find many papers explicitly studying this set, the structure and cardinality. I found one paper Division polynomials and multiplication formulae of Jacobian varieties of dimension 2 by N. Kanayama where he defines this set in Section 3.2.2.

Is it correct/wrong to call it generalization of torsion points?

I am interested in the cardinality of the set $$ J(C)_{\Theta}[n] \cap J(C)_{\Theta}[m], m, n \in \mathbb{Z}, m \neq n$$.