Are Reversal-bounded Multicounter Machines closed under reversal?
DCM is not closed under reversal.  I don't know of a reference, but it is easy to show it using the expressveness lemmata of Chiniforooshan, E., Daley, M., Ibarra, O.H., Kari, L., Seki, S.: One-reversal counter machines and multihead automata: revisited.

In particular, they state that for any DCM language $L$, there is a word $w$ such that $L \cap w\Sigma^*$ is a nontrivial regular language.  Now let $D$ be the following hinted version of the complement of the prefixes of the Dyck language:
$$D = \{a^nw_1w_2\cdots w_k \mid w_i \in \{[, ]\} \land |w_1w_2\cdots w_n|_] &gt; |w_1w_2\cdots w_n|_[\}.$$
In other words, $D$ is the complement of the Dyck where the number of $a$'s in the prefix gives a position witnessing the unbalancing of the parentheses.  Also, $(\{a\}^*)^{-1}D$ is the complement of the prefixes of the Dyck.  Note that $D$ is in DCM.  Now consider $D^R$ and a word $w \in \Sigma^* = \{a, [,]\}^*$.  Then either $D^R \cap w\Sigma^*$ is finite, or it is nonregular, hence $D^R \notin$ DCM.

I don't know of any interesting subclass of DCM closed under reversal.  (Ad placement!) However, we studied a model called Unambiguous constrained automata which is close but not comparable to DCM, and which enjoys closure under reversal --- constrained automata, or Parikh automata, themselves are equivalent to NCM in the nondeterministic case, and a strict subset of DCM in the deterministic case.