it sounds strange but symmetry is actually used a lot to study chaos.
in dynamical systems, chaos is more or less defined to say that “nearby things behave very differently from each other and go all over the place”. but there can also be an order to this unpredictability. one example would be the complex squaring map f(z) = z^2^ on the unit circle, which has the symmetry (-z)^2^ = z^2^, but this function is still chaotic (as you keep squaring certain numbers they can grow further apart).
i don’t know of a formal definition of chaos in algebra, but something that might perhaps sound a bit “chaotic” about polynomials is that there is no formula that gives the root of degree 5 polynomials. in other words they’re no analogue of the quadratic formula for polynomials with an x^5^ term. some have roots that can be solved for, and others don’t. the field of math that studies this is known as galois theory and it is all about studying symmetries.