I don't think this is a correct statement: "one flying off in one direction in time and the other flying off in the opposite direction in time" - we are talking about one particle making a U-turn in time, and because we experience time in only one direction, the phenomenon appears to us as birth...
There is an old weird theory that an antiparticle is actually a particle going back in time; when a pair is born "out of nowhere", it's in fact one particle making a U-turn in time.
Now, consider the expanding Universe. Let's say you are observing a nearby spinning galaxy drifting away from...