”upside down” could be translated as ”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.
BTW, it would be fun to see Diana Ross’ song ”Upside down” in Esperanto… :-)