Oh, I forgot to add to the part about mochi, but the Japanese when they look at the moon, instead of seeing a mans face they see a rabbit pounding mochi. It looks like this:

I got a book of Japanese folklore which has the story of how the rabbit came to be in the moon.

In the woods there lived 3 friends, a fox, a raccoon and a rabbit. While they were having dinner around their campfire the man on the moon saw them and wanted to find out who was the nicest of the 3.

So he came down from the moon and changed himself into an old begger man and approached them. He said, “I’m a poor old man and I’m starving, could you help me please?” The animals said, “oh you poor thing, let us find you something to eat.” And they ran off to find something for him.

The raccoon came back with an armload of nuts and berries, the fox caught a fish, but the rabbit couldn’t find anything and was sad. But then he had an idea and went and gathered firewood.

They all went back to the campfire and presented their finds. But the rabbit instead built up the fire as big as he could. The rabbit then said, “I couldn’t find anything for you to eat so I built up this fire and I’m going to throw myself in so you can eat me.”

Right as he was about to throw himself in the old man transformed back into the kami (god) and stopped the rabbit. The old man said, “Rabbit, you are the kindest and most generous of your friends, so I’m going to reward you by taking you to the moon with me.” And so they went to the moon and the man taught the rabbit how to make mochi so he’d have something to bring with him to the parties the kami throw.

Japanese Children’s Favorite Stories, Compiled by Florence Sakade, Tuttle Publishing, 1959