Mother Goose said it all.

The film's title was derived from a familiar, tongue-twisting Mother's Goose children's folk song (or nursery rhyme) called Vintery, Mintery, Cutery, Corn. The ones that fly east and west are diametrically opposed to each other and represent the two combatants in the film. The one that flies over the cuckoo's nest [the mental hospital filled with "cuckoo" patients] is the giant, 'deaf-mute' Chief:

Vintery, mintery, cutery, corn,
Apple seed and apple thorn;
Wire, briar, limber lock,
Three geese in a flock.
One flew east,
And one flew west,
And one flew over the cuckoo's nest.