She gave birth to the twins Phylacis (Phylacides) and Philandros (Philander). The Cretan tales (similar to the one about Romulus and Remus) are saying a goat suckled the twin-sons in the place named Elyros. Among Acacallis' children were also mentioned Cydon (the son of Apollo or Hermes), Miletos and Oaxos. King Minos, the father of Acacallis was not pleased with this situation and he punished his daughter by banishing her to Libya. There in Libya Acacallis gave birth to Garamant and it was also said that Garamant was the first man who was ever born.
Acacallis is named as the mother of the founders of the Western Cretan cities, but her influence (or the Minoan influence?) spread also over the eastern coast of Asia Minor and over the northern part of Libya. She was personally related to the city Tarrha where, as the myth is telling us, she met Apollo. Her twin-sons were associated with the city Elyros, which was a great city-state of the Classical Greek times, flourishing during the Roman and Christian periods. Her sons Cydon and Oaxos were considered as the founders of Cydonia (modern city of Chania) and Axos. They also said that Cydon was as an infant suckled by a bitch and this scene was depicted on the coins of Cydonia, where he later on became a hero. The name of the other brother Miletos is similar to the name of the ancient Cretan city Milatos, mentioned by Homer. According to the mythical story the inhabitants of this Cretan city under their young leader Miletos, son of Acacallis (or under Sarpedon, who was king Minos' brother), were the founders of the great Ionian city Miletos on the coast of Asia Minor. Acacallis' influence spread via her last son Garamant also to the northern part of Libya.
The myth about Acacallis carries some Roman influence, but the name Acacallis has a very old origin. C. Davaras assumed that this name has pre-Hellenic roots due to the reduplication of the first syllable and that it means "narcissus". This flower is actually flowering in Crete and it exists in its wild form as "the beach narcissus" between protected species. The name of Acacallis is used in a modern version as Akalle (or Akalli) for some places in Western Crete.