Not at all a physicist, but I think I can take a stab at this. I think so far we only see CPT (charge, parity & time reversal) symmetry as being a fully unbrok

Not at all a physicist, but I think I can take a stab at this. I think so far we only see CPT (charge, parity & time reversal) symmetry as being a fully unbroken symmetry of nature within the context of what you're asking. (I believe the weak force breaks a simple time reversal symmetry.) Time symmetry is typically synonymous with energy conservation in classical mechanics. It get's more complicated when you introduce quantum mechanics, and lorentz invariance from relativity. In essence though, there's probably an analogous invariant to energy conservation for CPT symmetry. Maybe this is just lorentz invariance for a quantum field theroy? Not quite sure. Implications of seeing that symmetry broken would be huge since the CPT theorem implies that a quantum field theory obeying lorentz invariance obeys a CPT symmetry. (I'm guessing by an application of Noether's theorem?) SO a violation of that would be either a violation of lorentz invariance (which would be VERY interesting) or some type of violation of the assumptions of quantum field theory. In essence, it would mean new physics!