Given a polynomial $P(x) \in \mathbb{Z}[x]$ and a ring $R$, call $P$ transitive with respect to $R$ if and only if for all $r \in R$, there exists a natural number $n$ such that $P^n(0)=r,$ where $P^n$ refers to the $n$-fold composite $$P^n = \underbrace{P \circ \cdots \circ P}_n.$$
Observe, for example, that the degree-$1$ polynomial $P(x) = x+a$ is transitive with respect to $\mathbb{Z}/r\mathbb{Z}$ if and only if $a$ and $r$ are coprime. I'm looking for a generalization of this statement.
Question. Is there a useful characterization of which polynomials $P(x) \in \mathbb{Z}[x]$ are transitive with respect to $\mathbb{Z}/r\mathbb{Z}$ generalizing the above comment, that's at least somewhat easier than simply repeatedly applying $P$?