Let's further specify that <PERSON> and <PERSON> start at the same location, and they later meet up and compare their viewpoints. <PERSON> just keeps going, while <PERSON> turns around and rushes to catch up.
<PERSON>'s perspective:
1. <PERSON>'s clock is running slow. She maintains steady speed for an hour, and calculates that <PERSON> has aged less than an hour.
2. <PERSON> turns around. To do so, she must accelerate rapidly towards <PERSON>. She calculates that, during her acceleration, <PERSON>'s clock ran fast. Very fast. How fast <PERSON>'s viewpoint thinks <PERSON>'s clock is running during her acceleration depends not just on how much <PERSON> accelerates, but also on how far <PERSON> is from her.
3. <PERSON> maintains steady speed approaching <PERSON>. <PERSON>'s clock is again running slow.
4. They meet and compare notes. <PERSON>'s clock running fast during <PERSON>'s acceleration actually made a greater difference than it running slow the rest of the time, so <PERSON> has aged more than <PERSON>.
<PERSON>'s perspective:
1. 844
<PERSON>'s clock is running slow. He maintains steady speed for over an hour, and calculates that <PERSON> has aged one hour.
2. <PERSON> turns around. This takes a short amount of time, and makes little difference.
3. <PERSON>'s clock, after momentarily running as fast as <PERSON>'s, is running slow again. He continues to maintain steady speed as <PERSON> catches up.
4. They meet and compare notes. <PERSON>'s acceleration had negligible effect on total time, so <PERSON>'s clock running slow the rest of the time means <PERSON> has aged less than <PERSON>.
End result, after meeting back up they both agree that <PERSON> aged less and <PERSON> aged more. They will even agree on the magnitude of the difference.
You might then ask why is <PERSON> the one that aged less? The reason for that is that she is the one that accelerated. When <PERSON> accelerated, she changed her reference frame. She was in one inertial reference frame before accelerating, and a different one after, and in a non-inertial frame while accelerating.
Calculating using the before-and-after frames, they have different definitions of what time is "now" at any location other than <PERSON>'s. Calculating using the acceleration frame, clocks run faster at locations she is accelerating towards, and slower or even backwards at locations she is accelerating away from, proportional to the distance between her and the clock.
She never actually sees a clock run backwards, however, because the effect scales at a rate that always allows the increase in travel time of the light from the clock to her to outweigh it - to overcome this limit and see a clock running backwards, <PERSON> would have to move faster than light.
Another followup question is what happens if neither <PERSON> nor <PERSON> accelerate, and they instead compare notes by sending messages? Answer: exact same thing, just with a message taking <PERSON>'s or <PERSON>'s place for part of the trip. They can't compare notes until the message from one is in the same location as the other, and achieving that will involve the message in question going through the acceleration frame transition that <PERSON> did in the original version. One way or another, the difference will get resolved by something accelerating before it can affect any physical interactions. 844