In the cosmological realm there seem to be no exact symmetries as with the quantum realm. Microscopic axes of symmetry appear to randomize on the large scale. Might even quanta be shown of complex structure? Can you think of any examples where outer space exhibits what would be considered precise symmetry? (I recall a basic symmetry to spacetime, but do not know whether it is local or global.) Even black holes, which obey simple geometries, are at least as entropic as the incoming particles at their boundary. The surface area of the black hole is a measure of its disorder, in turn proportional to the virtual Hawking radiation manifesting there. Then there are other "no hair" properties to consider. Could entropy itself relate to a symmetry?