This article does an amazing job of describing the various versions of the definition of "time" used in physics. The author limits his use of jargon, but defines them well when necessary. Highly recommended.
I wonder if the statement about Newton's second law (there exists an r(-t)) holds true when you include relativistic effects.
Time is reversible in general relativity, in the sense that if you replace t by -t everywhere the solution is still valid. I don't remember the full proof off the top of my head, and the argument I do remember is complicated.