Decidability of languages containing bitstrings and their corresponding ones' complements
Given a language $L$, you can create a language $L'$ which is equivalent in power and OCC:
$$
L' = \{ 0 x : x \in L \} \cup \{ 1 c_1(x) : x \in L \}.
$$
The two languages are recursively equivalent. This means that there is a computable reduction from $L$ to $L'$ and another one from $L'$ to $L$.

Using this construction, you can now answer your own question.