I searched for an operator !! in perlop, and not finding it have to believe it's a double application of !, essentially "Not-not". It seems to have the effect of making LHS the same as RHS, whether RHS is defined or not. That was torturous to say; do I have it right?

Almost, but not quite. It makes the LHS the truth of the RHS, not the value. See Bang bang in perlsecret - which is also the place to look for any other combination-operator idiomata you might happen upon.