Mr. West's birth date is revealed as July 2, 1842. Because West and Artemus Gordon were depicted as working for President Ulysses Grant, whose administration ran from 1869-1877, West would have been between 27 and 35 during The Wild Wild West (1965).
West's birthday is July 2, the same as the episode's co-writer John Kneubuhl.
The character Janus references the Roman god of the same name, who was known for having two faces.
Robert Conrad also played a character named (Milo) Janus in the Columbo episode An Exercise in Fatality.