If "time" is running backwards I dont see how that we would ever know, perhaps some clever physicist could devise an experiment to find out for sure.
Some theoreticians have suggested the reason the arrow of time runs from past to future is tied intimately to the expansion of space. Since Einstein special theory of relativity we have come to see that "time", space and speed are all relative. We now more correctly speak of "space-time" instead of just space.
So as the universe continues to expand time runs from past to present. Some have suggested that as the expansion slowly comes to a halt and if there is enough matter in the universe that it would reverse and we would be headed towards the "Big Crunch". At this time some speculate that the arrow of time would be reversed and "time" would go from future to past.
This reversal would mean that light would be going into stars instead of going out, and that thermal dynamics would be reverse as well as every thing else.
Our thoughts would also be reversed which would lead me to conclude that we would not know a reversal of time was in progress. I think that in our minds it would still seem that stars were "giving off light" instead of sucking it in, and that the laws of thermodynamics followed the same order.