It's time for a new constellation, and this one is entirely dedicated to two brothers. While there are many twins in Hellenic mythology--Artemis/Apollon, Iphikles/Hēraklēs, Amphion/Zethos, etc., this constellation is almost solely connected to one set of them: Kastor and Polideukes. In fact, the main stars of the constellation are named after them.

 

Hyginus briefly describes the constellation in his Astronomica, and focusses almost solely on the Dioscuri:

"These stars many astronomers have called Castor and Pollux. They say that of all brothers they were the most affectionate, not striving in rivalry for the leadership, nor acting without previous consultation. As a reward for their services of friendship, Jupiter [Zeus] is thought to have put them in the sky as well-known stars. Neptune [Poseidon], with like intention, has rewarded them for he gave them horses to ride, and power to aid shipwrecked men.
 
Those who speak of Castor and Pollux add this information, that Castor was slain in the town of Aphidnae, at the time when the Lacedaemonians were fighting the Athenians. Others say that when Lynceus and Idas were attacking Sparta, he perished there. Homer states that Pollux granted to his brother one half of his life, so that they shine on alternate days." [2.22]

The twins were born from Leda, Zeus, and mortal king Tyndareus. One of the twins was mortal, the other immortal, and both hatch from an egg. Accounts vary about who was the mortal one, and who was the immortal one; in some accounts they were even both mortal or immortal. During their lifetimes, the Dioscuri became great heroes, joining Iásōn in securing the Golden Fleece, and they both fought the Calydonian Boar.

I will discuss their mythology better another time, but for now, let me suffice in saying that eventually, they aspired to marry Phoebe and Hilaeira, who were already betrothed to cousins of the Dioscuri, a second set of twin brothers named Lynceus and Idas of Thebes, sons of Tyndareus's brother Aphareus. This sparked a feud between the cousins that left Polideukes the only one standing. He was given the choice by Zeus of spending all his time on Mount Olympus or giving half his immortality to his mortal brother. He opted for the latter, and thus the two divided their time between Olympos and the Underworld.

The Dioscuri were regarded as helpers of mankind and held to be patrons of travelers and sailors. Because they excelled in horseback riding and boxing, they were regarded as the patrons of athletes and athletic contests, and in all capacities, they had shrines throughout Hellas, and were worshipped well into the Roman era. As the twins lived on after death, they became emblems of the Eleusinian Mysteries, and were said to have been initiated into it during their lifetimes.

Hyginus also mentions a few other duos connected to the constellation:

"Others have called them Hercules and Apollo; some, even Triptolemus, whom we mentioned before, and Iasion, beloved of Ceres [Demeter] - both carried to the stars." [2.22]

Hēraklēs and Apollon earned this honor because, when Apollon commanded that Hēraklēs be sold into slavery to atone for the murder of his family, Hēraklēs became enraged and wrestled the Theos for the Delphic tripod. Their match was memorialized amongst the stars as the constellation Gemini.

Triptolemos and Iasion are the last candidates. The two were favorites of the Goddess Demeter. Triptolemos was a hero who first instructed mankind in the art of agriculture, while Iasion was said to be Her lover on the island of Samothrace. Demeter must have immortalized them out of love.

Gemini is visible at latitudes between +90° and −60°, and best visible at 21:00 (9 p.m.) during the month of February.