Symbols mean different things to different people. They resonate differently in different
minds. A favorite phrase in mythological stories is: "According to some..."
The Wikipedia article on the labrys is here.
The feminist Amazonation site says this1.
In the Oresteia, Aeschylus tells us that the androgynous Clytemnestra used a battle-axe and net to kill Agamemnon on his return from Troy.
For me, Clytemnestra is the Queen of the Night - a time-goddess, related to Demeter‘s dark other self, Persephone. Demeter and Persephone are the goddesses of fruitful and unfruitful seasons respectively. Clytemnestra also had another half - Helen, the beautiful Queen of the Day. Agamemnon is the King of the Day, the Sun-King, who rises in the east, and meets his death in the west.
Aeschylus represents the night by women and the day by men, strengthening the idea of an unbounded cycle by starting the Oresteia after the death of Iphigenia. Clytemnestra kills Agamemnon because he killed her daughter. Consequently, she is killed by her son. Whenever time moves on, the sky turns blazing red. Eventually however, the Oresteia ends happily with the triumph of Athena.
The battle-axe, one of Clytemnestra’s attributes, was a symbol of the Demeter cult. It is obviously androgynous too — it has a shaft, and is not just an axe-head! The horizontal time-axis is birth and death, while the vertical axis is conception. Both dimensions together represent past, present and future, while in an invisible third dimension there is a parallel symbol with a life of its own: Transcendence of time by exuberant, triumphant love. Is this a forerunner of the Christian cross? The labrys is above all a symbol of time - like father time's scythe.
Clytemnestra’s other attribute, the net, is related to hunting. Iphigenia is related to Artemis, goddess of hunting, and to Hecate, a reclusive goddess who can see past, present and future... Hecate was the arch witch conjoured up by the three witches in Macbeth. Some say that she was a sister of Demeter and Persephone...
The original Amazonation source is here
(December 2008). Some relevant extracts are:
The labrys is a symbol of the female labia at the entrance of the womb and the butterfly, which is connected with rebirth. The double axe is also associated with the even more ancient hourglass figure of the Goddess.
This symbol also marked the entrance to Goddess sanctuaries.
The two heads symbolize the waxing and waning Moons. The labrys design is found on matriarchal murals and mosaics, pottery, seals, and amulets. It was exclusively a symbol of the Great Goddess, until part of its symbolism was later transferred to the Nordic god Thor. - DJ Conway
The labrys or double-bladed axe stood for the Amazons and their Goddess under several of Her names: Artemis, Gaea, Rhea, Demeter. Perhaps originally a battle axe, it became a ceremonial scepter in Crete and at the Goddess's oldest adopted shrine, Delphi. - B. Walker
The labrys (labyris - labris - labrus) is an ancient Minoan symbol that looks like a double axe and was quite common on the island of Crete. Scholars have reason to believe that the symbol actually represents a butterfly, a symbol of transformation. This symbol was most likely the one at the end of a wand that was supposedly used by a goddess (Circe, Athena). Later this wand was actually transformed into an axe that was used as a weapon by the Amazons, a matriarchal warrior society...