It used to be true (except the part about showing it off), as the Russians had the skull and performed an autopsy. But in 1970 the body was exhumed and burned, the ashes dumped in the Elbe river.
Hitler's jaw does still exist as it had been removed and taken to Moscow in 1945. They also have a skull fragment which may or may not belong to Hitler; it was found in the hole in which he had been buried in 1946 but has not been authenticated.
