Douglas Harper's Etymology Dictionary
pushmi-pullyu
fictional two-headed mammal, from "Dr. Dolittle" (1922), coined by Hugh Lofting from the expressions push me, pull you. Popularized by the 1967 film version of the book.
Wiktionary
pushmi-pullyu
a. 1 conflicting or contradictory. 2 in teleosemantics, being a representation with both indicative and imperative content. n. a fictional animal with two heads at opposing ends of its body, in Hugh Lofting's ''The Story of Doctor Dolittle''.