Since the height of the Southern Cross is the primary latitude clue for Hawai'i, and since measuring its height depends on the visibility of the horizon, which in turn depends on the moon lighting the horizon, you can see the important relationship between these stars and the moon. So the Southern Cross is "Cared for by the Moon."

Also, the line dividing the light and dark sides of the moon points north and south, and the Southern Cross also points to the south, so that is another connection between the moon and the Southern Cross.