The speaker pictures a hare out on a white, snowy hill, free and wild, and then sets that image against himself stuck at the office. The question "which is which" blurs the two: is he the hare, or is the hare him? The title plays on "hare-brained," a word for foolish or crazy behavior, and the poem asks who deserves that label.
The speaker admits he "may alter" his opinion about who is really mad. At first the hare seems like the crazy one, but the poem tilts the other way. The office is "squeezing," a word that makes work feel tight and suffocating, and the speaker ends up "out of order," like a broken machine or someone who doesn't fit the rules. The mood is wry and restless. There's humor in the wordplay, but also a quiet longing for open space, and a suspicion that ordinary routine may be the stranger kind of madness.