So if someone who is “unhinged” calls you “unhinged”,  does that make you “hinged”?

The term “unhinged” means:

mentally unbalanced; deranged:
“the violent acts of unhinged minds”
So it is safe to say they are probably not using logic and reason.  Now I know logic and reason are not super fashionable right now, especially in the political realm, but humor me.  If someone not in full possession of their faculties calls you “unhinged” it means they are not using logic and reason so rather than implying that you are in fact “hinged” it just means their opinion is lacking is gravitas or if that word is too fancy it means their opinion is “without serious merit”.  Still too fancy?  OK, you got it, “worthless” it is.
We conclude therefore that no, you are not necessarily “hinged.”  You might in fact be “hinged” (aka “sane) but that not an express conclusion.