Thanks to Odder, all good info.
Yes, we believe that Lorentz invariance holds to about 1 part in 10^17, and even then, we're (probably) measuring the limits of the experimental setup; our technological abilities. That is to say, Einstein's theories of relativity (both special and general), have never been violated, to the best of humans' ability to measure such a thing.
There is, in fact, a theoretical particle that moves backwards in time, called a tachyon, but if they do exist, there is no known interaction with us, or any baryonic matter, for that.. err, matter. Here's a classic joke among particle physicists:
If you are an educated physicist please turn OFF everything you have learned just for a minute, and just let your mind wander a bit into the unknown.
I'm afraid I can't do that, Dave. This is in the realm of known physics.
You are correct in thinking that a grand unifying theory of physics may require reformulating everything that we know in a completely novel set of axioms. String theory is one attempt at this, though it doesn't make testable predictions that would reveal its superiority over existing theories. And I'm personally under the impression that there will never be another singular person (another "Einstein") capable of bestowing us with a grand unifying theory; it will probably require teams of people working iteratively over several generations. Things have just gotten too damn complicated. But I hope I'm wrong.
It's healthy to muse on things like this, though, I don't want to discourage that. I can guarantee that I would look like Hubski's ultimate fool if I were start bumbling on about economics, poetry, politics, or anything that isn't physics. Apparently, to become an educated physicist these days, it requires hardly knowing anything else, at least until you get the PhD. And I've already cried about it, my tear ducts are drier than the Sahara, at this point.