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... I think it is even easier and certainly more concrete to think about it in terms of Doppler shift. Both twins SEE (literally SEE) the other twin aging more slowly as they move away from each other because of the Doppler shift and both SEE the other aging more quickly as they move towards each other. The traveling twin SEES equal periods of slower and faster aging because the Doppler shift changes for her immediately upon turning around. In contrast, the stay at home twin sees a much longer period of slower aging, because he doesn't see the Doppler shift change until the light from the turnaround event gets to him. Thus, the traveling twin is younger when she returns.
See http://www.csupomona.edu/~ajm/materials/twinparadox.html for a more complete explanation and spacetime diagrams.