I just wanted add (similar to what I think Severian is pointing to) that the "paradox" could happen in different gravity situations too. If one twin stayed on Earth, and the other were instantly beamed to a neutron star (and could survive there), stayed there for a year, and then was instantly beamed back to Earth, the twins would notice the neutron traveler had aged less, and that his watch was behind the Earth twin's watch.
Relativity can be understood better if we look at the history of the universe. What we call "time" is simply our observation of the universe changing at a certain rate. That change is toward disorganization, so time is really the rate of entropic change. The universe started out tightly compacted, it is now flying apart. The balance between this density and expansion is moderated by gravity, which is also reproduced when one accelerates. What gravity and acceleration do, in a sense, is create anti-expansion and is therefore anti-entropic. It slows everything down, sends it back toward its originating direction of density. So the rate of entropy of clocks, body aging, and everything else in a frame of reference are affected.
A question I asked at another science site once was if the traveling twin, Bill, would notice in any way. Let's say Bill spends 35 years on Earth, takes off in his space ship, and observes his clock register the passing of five years. When he gets back to Earth, he finds his twin has aged 15 years. Would Bill feel like those were the longest five years he'd ever experienced? Remember Bill had had 35 years in Earth's frame of reference. He would have that at least to compare to his space ship frame of reference.
All the physicalist science types insisted there would be absolutely no way to tell in terms of measurement. I agreed. But what I wanted to know was if Bill might feel it. Really I was posing a question about the nature of consciousness. If consciousness does emerge from the physical processes of the brain, then Bill probably wouldn't notice. But if consciousness is not physical but, for instance, some quality associated with the brain, then it might have its own internal frame of "feeling" reference which might sense it had lived through an awfully long five years.