The Moon & Cancer
The tale of Cancer in Greek myth most often corresponds with the crab in the story of the 12 Labors of Hercules. While Hercules is busy fighting the Hydra, the goddess Hera sends a crab to distract him. This pretty much doesn't work because Hercules steps on the crab, but apparently Hera felt bad so she put the crab in the heavens.
I am not enamored of this tale for Cancer. The crab may be a fitting symbol, but not because of a useless sacrifice. While giving up one's life for a cause is a heroic ideal, I don't think that ideal is represented in this story, and it misses the point and beauty of Cancer. Consider these thoughts:
Cancer has been called the 'Gate of Men' where souls are born into human form (Capricorn being the opposite-where souls leave).
In Egyptian mythology, the scarab was associated with the constellation we know as Cancer, which they called Scarabaeus. The scarab is a beetle that rolls its dung into balls and buries its eggs there. The hieroglyph of a scarab has two meanings: that of the actual beetle itself, and of the cyclical nature of life. The scarab is a symbol of immortality or rebirth.
The Egyptian god, Khepri (whose name means 'to come into being') was associated with the scarab because Khepri pushes the sun across the sky, and is associated with the dawning of the Sun each morning (as the sun 'comes into being').
Arguably the most notable feature about the constellation is the beehive cluster, named Praesepe, which is a Latin word meaning manger or crib.
The stories I came across all linked Cancer back to the idea of birth, of life, whether it's the creation of life or the protection of life, as the crab's shell protects it from harm. The essence of Cancer is a nurturer, a protector, one who wears a shell only to protect and explore the vulnerability within, and those she takes into that safe space. Cancer is a lifegiver in all symbolic and literal forms. Cancer's desire to nurture can go awry when it sacrifices itself, giving too much without receiving. Perhaps the crab in the Greek story should have told Hera to find another sucker!
Earth's moon is one of the oldest symbols known to us, pre-dating the written word. Every culture has myth about the moon, represented in male and female deities, or even as a world on which a deity lives. The Greek goddesses Hecate and Artemis are often connected with the Moon, as well as the Roman goddess Diana (of the hunt). In the Chinese culture the Moon doesn't represent a deity, but a place where immortals live, such as the deity Chang'e.
Even more prominent is the symbolism of the moon's phases: as it waxes and wanes from the new moon to full and back again. Every culture seems to, in some way, associate the moon with all things that change and shift. The idea of a Triple Goddess represented by the waxing crescent, the full moon, and the waning crescent is prominent in the Wiccan and Neo-Pagan religions as three phases of a woman's life: the Maiden, the Mother (with her full, round bully like the full moon), and the Crone. This is yet another example of the representation of the cycle of life wrapped up often in Cancerian and lunar symbolism.