well forwards time travell is well established. My favourite is the muons hitting the Earth from cosmic rays
http://www2.slac.stanford.edu/vvc/cosmicrays/c ratmos.html
(If anybody doesn't know all about this anyway).
The lack of evidence is interesting, after all you might think there was no evidence for radio wave 200 years ago - we didn't have the technology.
As many of these things revolve around extreme cases that's not too surprising.
So I don't think a lack of evidence should push us one way or another.
You could interpret the fact that relativity doesn't forbid it as evidence that it's possible but Gravity is very poorly understood in extreme circumstances so I don't think that helps us either.
Should we ever manage a theory of quantum Gravity and that too allows for backwards time travel It'd be much stronger evidence.
Of course Richard Feynmann famously pointed out that in some senses a positron can be seen as an electron moving backwards in time. That is generally seen as just a mathematical device rather than having any real foundation in the physical world. But if I wanted to invent a time telephone I think I'd be looking at anti-matter rather carefully