QuickCheck: Is it true that Native Americans also have a 'Moon Rabbit'?

The Maya Moon goddess is frequently depicted with her rabbit companion.

CULTURES all around the world have various stories and legends about the moon and many have to do with how it looks or what people think they see when they look at it. Like an international Rorschach test, different cultures have come up with various stories describing the dark markings on the near side of the moon (the side that faces us). They've seen faces, people and even toads but the pareidolia from our part of the world has a distinct “bunny” flavour as folks in East Asia, as well as South-East Asia, say they see a rabbit.

However, East Asia isn't the only place that sees rabbits when they look at the lunar maria and highlands of the moon. Is it true that Native Americans also have tales of a “Moon Rabbit”?