I think I'm trying to say something about global reference frames.
If I understand correctly, the "right" answer to your questions is:
"There is no coordinate system which contains both the westbound and the eastbound twins' paths in their entirety."
Consider this: in a real special relativistic reference frame, the westbound twin should never meet the eastbound twin again; the eastbound twin keeps going east, east, and more east.
However, we've all played enough Asteroids to know how to patch this up; draw two vertical lines which are supposed to represent the same place in the universe and say that anyone that goes past the line on the right wraps around to the line on the left.
The problem is that Asteroids has mislead us! In the simplistic universe I described, those lines are generally not supposed to be vertical lines; they should be slanted lines... so when you "wrap around", you change
both your spatial position and your temporal position.
(equivalently, you can draw the lines vertical, but when you wrap around, you are also shifted... so, for example, the event 100 light-years to the east could be the very same as the event 10 years in the future)
The right answer, I guess, is that you're supposed to use two different coordinate systems, one for, say, the first two-thirds of the trip and one for the last two-thirds of the trip, and get the right coordinate transformations in the region where they overlap.
And I'm saying all of this because I don't know what you mean in your last post.
