What sort of experiment would directly test time reversal invariance? I guess the title says it all: how could/would you experimentally test whether our universe is truly time reversal invariant, without relying on the CPT theorem? What experiments have been proposed to check this? Have any of them been performed?
I know that there are indirect tests of time reversal invariance by observing CP violation, in the decay $K_L \to 2\pi$ for example. Then if you assume that the necessary conditions for CPT symmetry are satisfied in our universe, that means there must be time reversal symmetry violation as well. But I'm curious about ways to test time reversal invariance without relying on CPT.
Basically, how could we distinguish between the Standard Model, which predicts T violation, and some hypothetical other theory that matches current experimental results as well as the SM, but in which CPT symmetry does not hold?
 A: At research level, you might be interested in the PDG review on conservation laws. Also, the review about CPT invariance gives information about tests of CPT violation in neutral kaons, at Phys. Lett. B 237, 303 (1990), Phys. Rev. D 67, 012005 (2003) and Phys. Rev. Lett. 74, 4376 (1995) for CPT violations, and at Phys. Lett. B 444, 43 (1998) and Phys. Rev. Lett. 83, 911 (1999) for CP violations.
Note that CP violation itself is still an active area of research (particularly at Belle and LHCb), since we don't know definitively how many systems display it and whether there's a deeper explanation for it.
A: There are numerous research groups engaged in a search for an electric dipole moment of the electron, which, if it exists, would violate time-reversal symmetry. You can see this because any dipole moment the electron might have would need to be parallel to the spin (or anti-parallel). When you reverse time, the spin necessarily flips, but the electric dipole moment would not change, so the relative orientation of the two would change. That's the best example of a T-violating phenomenon that I know of.
At the risk of unseemly self-promotion, I wrote an article on edm searches for Physics World last year. You need to register to read the whole thing, but it's free.
A: As an update on this old thread, the 2015 version of the Particle Data Group review on tests of conservation laws (the 2009 version of which was rightly pointed to by invisiblerhino) has an interesting update:

The BABAR experiment has reported the first direct observation of $T$ violation in the $B$ system. The measured $T$-violating parameters in the time evolution of the neutral $B$ mesons are $∆S^+_T = −1.37±0.15$ and $∆S^−_T = 1.17±0.21$, with a significance of $14σ$ [4]. This observation of $T$ violation, with exchange of initial and final states of the neutral $B$, was made possible in a $B$-factory using the Einstein-Podolsky-Rosen Entanglement of the two $B$'s produced in the decay of the $\Upsilon(4S)$ and the two time-ordered decays of the $B$'s as filtering measurements of the meson state [5].

Pointing to the reference

[4] J.P. Lees et al., Observation of Time-Reversal Violation in the $B^0$ Meson System, Phys. Rev. Lett. 109, 211801 (2012), arXiv:1207.5832.

which has pretty much what it says. For an entry-level explanation of that paper, the APS Physics Viewpoint: Particle Decays Point to an Arrow of Time is probably a good place to start. That article probably does a much better job than I could at explaining the particulars, but I'll note here that, with a $14σ$ significance, this experiment does seem to mean that

the long wait for an unequivocal time-reversal violation in particle physics is finally over.

A: The C, P and T symmetries came from the equations due to Schrodinger, Klein-Gordon and Dirac. These partial differential equations depend on time and position. By changing the sign of time and position, these equations remained unchanged, so here is the origin of P and T. The C symmetry resulted originally from Dirac's equation. Since these equations are very good at describing the behaviors of particles, we can safely say that C, P and T came from observations.
After the discovery of the weak interaction, C and P turned out to be violated, and the hope was that their combination CP is still preserved. After the discovery of the violation of CP, the hope moved to CPT. So, the experimental data said that each of C, P, T is respected when no weak interaction takes place, otherwise we should be happy with the combined CPT.
So, I think that the answer is that the experimental data told us that the universe is T-invariant, then when the weak interaction was involved, it told us that CP is violated, and from CPT that T is violated. And CPT is not specific to the standard model, it results from the Lorentz invariance and the fact that the energy is bounded for below.
A presumable CPT violation would imply the violation of the Lorentz invariance (http://en.wikipedia.org/wiki/CPT_symmetry#CPT_violation). So, if an alternative theory wants to distinguish itself by experiment, it should predict violations of the Lorentz symmetry. The theory itself will say the regimes in which this violation occurs (I bet that it will be at the Plank scale!). In this case, probably the Lorentz violation will imply with ease the CPT violation.
I find it very unlikely a violation of the Lorentz invariance, because this principle is (probably) the most ubiquitous principle in Physics, but who knows...
