Dominations under oracles which is closed under complement?
For a given choice of complexity classes $\mathsf{C}$ and $\mathsf{O}$, by which we mean sets of languages and nothing else, the class $\mathsf{C}^{\mathsf{O}}$ is not well-defined: we need the definition of $\mathsf{C}$ to determine how oracle queries are defined.  Sometimes even then there could be ambiguities that need to be resolved, or the definition might not provide a reasonable notion of an oracle query at all.  (Attaching oracles to space-bounded complexity classes is one example in the first category, where the relativized classes we obtain are very sensitive to the exact capabilities of the oracle tape and whether we consider that it contributes to our space bounds.)

So, I don't think you can possibly say anything about $\mathsf{C}^{\mathsf{O}}$ without using the definitions of the classes, because if you don't use the definition of $\mathsf{C}$ you aren't working with a well-defined mathematical object.  

If I have misunderstood your question, please clarify what sort of properties of $\mathsf{C}$ and $\mathsf{O}$ you permit to be used.