There is at least one philosopher before Plato and he is Anaximander. There are many passages in his works that relate to the concept of symmetry:
The basic elements of nature (water, air, fire, earth) which the first
Greek philosophers believed that constituted the universe represent in
fact the primordial forces of previous thought. Their collision
produced what the mythical tradition had called cosmic harmony.
Anaximander claimed that the cosmic order is not monarchic but
geometric and this causes the equilibrium of the earth which is lying
in the centre of the universe. This is the projection on nature of a
new political order and a new space organized around a centre which is
the static point of the system in the society as in nature.
In this space there is isonomy (equal rights) and all the forces are
symmetrical and transferrable.
But his most famous reference to symmetry is his statement that:"..the round earth is situated in the midst of empty space but it does not fall...because there is no preferred direction for it to fall in!
Anaximander was the first to conceive a mechanical model of the world.
In his model, the Earth floats very still in the centre of the
infinite, not supported by anything. It remains "in the same place
because of its indifference",... Anaximander's realization that the Earth
floats free without falling and does not need to be resting on
something has been indicated by many as the first cosmological
revolution and the starting point of scientific thinking. Karl
Popper calls this idea "one of the boldest, most revolutionary, and
most portentous ideas in the whole history of human thinking."
Such a model allowed the concept that celestial bodies could pass
under the Earth, opening the way to Greek astronomy.