I've been thinking about transformations on $NP$-complete problems that produce languages known to be in $P$. However, I can't seem to find an example of two $NP$-complete languages whose union is in $P$. I would imagine that such a pair exists (perhaps something like "every object has exactly one of two properties, but it's $NP$-complete to determine whether any given object has one of those properties"), though I don't know anything of this sort.
Are such pairs of languages known to exist? Or is their existence an open problem?