# Decidability of bitstring languages with no ones' complements

For any string $s \in \{0,1\}^*$, let the ones' complement of $s$, denoted $1_c(s)$, be the string obtained by flipping all bits in $s$. For example, $1_c(001100) = 110011$. Now call a language $L \subset \{0,1\}^*$ ones' complement asymmetric or OCA for short, if for every string $s \in \{0,1\}^+$, precisely one of $s$ and $1_c(s)$ is in $L$.

I want to come up with a mapping reduction from a language $L \subset \{0,1\}^*$ to an OCA language (and vice-versa) so as to prove the existence of decidable/recognizable and undecidable/co-recognizable and undecidable, etc., OCA languages. My problem is especially that we require every string in $\{0,1\}^+$ to be such that either it or its ones' complement appears in any OCA language. This seems particularly problematic if $L$ lacks strings of a particular length, meaning it isn't obvious how we'd go about constructing an OCA language from $L$.