The title card "What happens in Cygnus X-1, stays in Cygnus X-1" is correct. Cygnus X-1 is a black hole from which nothing escapes.
The Binary code on the door to Bender and Fry's apartment translates to "$."
The Stargate SG-1 (1997) episode Holiday (1999) had the same can't-swap-back problem with the same circuitous solution.
The episode title is an homage to the 1894 Anthony Hope adventure novel, "The Prisoner of Zenda," in which a vacationing Englishman is reluctantly recruited to impersonate the indisposed King of Ruritania. In this episode however, Bender switches bodies with the Emperor to impersonate him, and with more nefarious intentions.
The mind-swap plot is similar to Bodyswap (1989). Including having other characters voices leave another's mouth and another gorging themself.