Time reversal invariance
The reversibility of the laws of physics implies that given the state of a physical system at a particular time, it is always possible to work out uniquely both its future and its past. Time reversal invariance would further imply that the rules for going in each direction should be identical. To a very good approximation this appears to be true, but it turns out that in certain esoteric particle physics processes small deviations have been found. In particular, it was discovered in 1964 that the decay of the K0 particle violated time reversal invariance at the level of about one part in a thousand. In current theories, this effect is not attributed any particularly fundamental origin, and is just assumed to be associated with the arbitrary setting of certain parameters. K0 decay was for a long time the only example of time reversal violation that had explicitly been seen, although recently examples in B particle decays have probably also been seen. It also turns out that the only current viable theories of the apparent preponderance of matter over antimatter in the universe are based on the idea that a small amount of time reversal violation occurred in the decays of certain very massive particles in the very early universe.
The basic formalism used for particle physics assumes not only reversibility, but also so-called CPT invariance. This means that same rules should apply if one not only reverses the direction of time (T), but also simultaneously inverts all spatial coordinates (P) and conjugates all charges (C), replacing particles by antiparticles. In a certain mathematical sense, CPT invariance can be viewed as a generalization of relativistic invariance: with a speed faster than light, something close to an ordinary relativistic transformation is a CPT transformation.
Originally it was assumed that C, P and T would all separately be invariances, as they are in classical mechanics. But in 1957 it was discovered that in radioactive beta decay, C and P are in a sense each maximally violated: among other things, the correlation between spin and motion direction is exactly opposite for neutrinos and for antineutrinos that are emitted. Despite this, it was still assumed that CP and T would be true invariances. But in 1964 these too were found to be violated. Starting with a pure beam of K0 particles, it turns out that quantum mechanical mixing processes lead after about 10-8 seconds to a certain mixture of K0 particles—the antiparticles of the K0. And what effectively happens is that the amount of mixing differs by about 0.1% in the positive and negative time directions. (What is actually observed is a small probability for the long-lived component of a K0 beam to decay into two rather than three pions. Some analysis is required to connect this with T violation.) Particle physics experiments so far support exact CPT invariance. Simple models of gravity potentially suggest CPT violation (as a consequence of deviations from pure special relativistic invariance), but such effects tend to disappear when the models are refined.