If clock B then also enters “dilated time” (together again with clock A) it has gone from September 26 2013 back to September 25 2013 (even though it reads September 26 2013).
I don't know what it means to "enter" dilated time. They both (clock A and clock B) arrive at the same point in space time having traversed different amounts of proper time in order to get there. However, there is no section on the worldline of either clock for which the proper time is negative.
I think I finally understand what you are saying: you have two clocks, one reads september 25, the other 26. We say that the one that reads 25 has travelled into the future by virtue of reading 25 while encountering a world in which clocks agree that it's actually 26. But it's equally valid to say that those other clocks have travelled to the past by virtue of reading 26 when encountering a clock that reads 25.
But here's the thing: That's not what I'm saying. I, at least, am saying that
everyone is travelling forward in time. You can see this easily with a space-time diagram. The amount of time that different people experience between space-time events can vary. That's it. Moreover, september 25 on clock A is not necessarily in the past of september 26 on clock B. In fact, in this scenario they are in each other's present. However, september
24 on clock A certainly is in the past of september 25 on clock B.
Some people say we are travelling forward in time at one second/second. That's of course, a relative statement. Relative to a clock somewhere (anywhere) on earth I am moving forward through time at one second (on my clock)/second (on that other clock). If I were to go away from the earth and come back, I could look at my journey and find that I had, over that time, travelled forward at an average of 1/10th second(on my clock)/second(on an earth based clock). Someone on earth could, during that same period, say that, relative to my clock, they travelled forward in time at a rate of 10 seconds (on their clock)/second (on my clock). At no point will either of us have a negative reading.