An imaginary creature resembling a llama or antelope, but with a head at either end of the body, pointing away from the torso, so that the creature always faces in two directions at once. Hence in extended and allusive use.
Widely popularized by the film version of Doctor Dolittle (1967).
1920s. From push + me + pull + you, with various arbitrary respellings of the pronoun forms. Coined by Hugh Lofting in Doctor Dolittle.