”upside down” could be translated as ”renversita”renversita. PIV gives some examples, such as: li renversis la vinbotelon, por ĝui ĝis la lasta guto ”he turned the wine bottle upside-down in order to enjoy even the last drop”.
I’m not sure about ”inside out”, though. I guess it could be reversita – PIV mentions ”reversi” in the context of re-sewing clothes with the inside out.
BTW, it would be fun to see Diana Ross’ song ”Upside down” in Esperanto… :-)