On a surface embedded in Euclidian space, where the metric signature is all positive, it is possible for a particle traveling along a geodesic to encounter a curved bit and then get turned around so that it heads in the opposite direction.
Could there be a manifold that did the same turning-around trick, but in a timelike direction? Would this create particles for which increasing proper time corresponded to decreasing coordinate time?
Yes, this happens when you enter or exit the inner Cauchy horizon of a rotating Kerr black hole at the right angle, see Madore's article and Hamilton's penrose diagram; this is sometimes referred to as the Carter time machine.
Inside the inner horizon it is even possible to meet an other observer who fell through at an other angle so his proper time is going in the same direction as the coordinate time, while your proper time goes in the opposite direction, which would have weird effects since not only would you hear the other observer talk backwards, but also he would remember things that are in your future while what's in your past is in his future.
That would be even weirder if you chose to blow yourself up, then the other observer would observe an implosion that would create rather than destroy you, and a bunch of other paradox thought experiments would become possible.
The only problem is that when you go through the Cauchy horizon, you get hit by an infinite blueshifted signal (in the moment when you cross that horizon, you go to future infinity and back in an infinitesimal amount of proper time), see Dafermos's lecture.
