Gods or goddesses. Or notable characters from any myth. All cultures are welcomed.
Well, the Maya Hero Twins are usually portrayed as two male twins, but the Maya had a more fluid sense of gender/sexual identity for their deities. One of the Hero Twins is often given feminine characteristics. Indeed, at the end of the Popol Vuh, the Hero Twins become the sun and the moon respectively. In most other sources on Maya religion, the moon is portrayed as a goddess, not a god. Additionally, the name of the twin that becomes the moon is usually rendered with some form of the prefix Ix (pronounced eesh) in front of their name, which is the female prefix. The most common rendition of his name is Xbalanque (i.e. Ixbalanque). The other alternative is that it is a variation on the word yax meaning precious or blue/green. There are some other indicators, but those are the two most obvious. You get similar sort of gender-bending with other Maya deities. Certain places in the codices (particularly the Dresden Codex) there are obviously male deities with breasts. For instance, the flower god.
I think the eldest two of the Children of Lir might count, or might not. The younger brothers are definitely twins.
I was just reading this in one of Eileen O'Faolain's anthologies, so I'm not sure if it's "ancient" enough for you .
There are always four children, an older daughter and son, and younger twins, and they're cursed to spend a long time as swans before being allowed to resume human shape. In O'Faolain's telling, they're swans for 300 years in one location, 300 in another, and then 300 in a third, and once they return to human form they nearly instantly age and die, but with enough time to convert to Christianity and to tell a priest how to bury them - with the daughter in the middle, the oldest brother before her and the two younger ones under each of her arms, just as they slept to keep warm when they were swans.
In this telling and in this telling, Fingula (the daughter) and Aodh (the eldest brother) are twins as well as their younger two brothers, Fiachra and Conn.
In O'Faolain, I think it's just stated that there are two older siblings and then same-sex twins, but I'm not positive.
Looking for other versions of the story turned up the "Divine Twins" entry in the Encyclopedia of Indo-European Culture, which might also be useful. It's meaty.