Let's say I have a decision problem $D$ and its complement $D'$. I know D is poly-time reducible to $D'$ (its complement). Furthermore, I know $D$ is NP-complete. What is the strongest statement I could possibly make about this kind of relationship?
If an NP-complete problem is reducible to its complement then NP=coNP (why?). Conversely, if NP=coNP then every NP-complete problem is reducible to its complement (why?).