Simultaneity (or lack thereof) is a terrific tool for understanding many of the paradoxes associated with SR. And, if I am to be thorough, simultaneity must be considered for all SR events between separate frames of reference. Let's re-visit the twin paradox (John travels out 12 hours at 60% the speed of light and returns at the same speed). Basically, there are three frames of reference to consider. First, the twins are on the earth with no relative velocity between them. Second, John embarks on the outgoing leg of his trip. Thirdly, John (after instantaneously turning around) embarks on his return leg of his trip. I am using the same example as before, except I am using numbers from the Lorentz Transforms as opposed to the Relativistic Doppler Shift to explain the observed phenomena.
Hunter and John each agree on everything they observe. This should be easy to understand since there is no relative velocity between the two twins. They are in motion together.
John travels out 12 hours by his clock. With the two postulates in mind, we realize that Hunter observes time dilation for John's outgoing trip. Thus, if John records 12 hours, Hunter will record 15 hours. Remember that at 60% the speed of light, the time dilation will be 80%. Therefore, if John records his time to be 12 hours, this is 80% of what Hunter records - 15 hours. But what does John observe for Hunter's time? He observes the time dilation as affecting Hunter; therefore, he measures his trip to be 12 hours, but he observes 9.6 hours (80% of his clock's time) for Hunter's time.
2nd frame totals:
Hunter measures his time to be 15 hours, but John's time to be 12 hours. John measures his time to be 12 hours, but Hunter's time to be 9.6 hours.
Obviously, the event, which is the end of the outgoing trip, is not simultaneous. John thinks Hunter's time is 9.6 hours but Hunter thinks his time is 15 hours. On top of that, they both think that John's time is 12 hours, which doesn't agree with either of the first two times.
We'll look at the results of this scenario in the next section.