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Let us focus on what Stella and Terence actually see with their own eyes. (Just to emphasize that we're talking about direct observation here, I'll put the verb "see" and its brothers in the HTML strong font throughout this section.) To make things interesting, we'll equip them with unbelievably powerful telescopes, so each twin can watch the other's clock throughout the trip. If each twin saw the other clock run slow throughout the trip, then we would have a contradiction. But this is not what they see.
Just in case it's too hard to read the clock hands through the telescope, we'll add a flash unit to each clock, set to flash once a second. You might guess at first that Terence sees Stella's clock flashing once every 7 seconds (with the time dilation factor we've chosen) and vice versa. Not so! On the Outbound Leg, Terence sees a flash rate of approximately one flash per 14 seconds; on the Inbound Leg, he sees her clock going at about 14 flashes per second. That is, he sees it running fast! Stella sees the same behavior in Terence's clock.
What gives? Well, the section title gave it away: just replace the words "flashes per second" with "cycles per second", and you'll recognize the familiar Doppler shift at work. The regular pulses are redshifted to lower frequencies during the Outbound Leg, and blueshifted to higher frequencies during the Inbound Leg. (I invite you to consider laser-based clocks instead of flash units, for added techno-jazz.)
The Doppler shift factors I gave (1/14 and 14/1) come from the relativistic Doppler formula. The relativistic formula takes into account both the "delay through distance" effect of the non-relativistic formula, and the relativistic time dilation. In other words, Terence computes that Stella's clock is really running slow by a factor of about 7 the whole time, but he sees it running fast during the Inbound Leg because each flash has a shorter distance to travel. And Stella computes the same for Terence.
All well and good, but this discussion at first just seems to sharpen the paradox! Stella sees what Terence sees: a slow clock on the Outbound Leg, a fast clock on the Inbound Leg. Whence comes the asymmetry between Stella and Terence?
Answer: in the duration of the Inbound and Outbound Legs, as seen. For Stella, each Leg takes about a year. Terence maintains that Stella's turnaround takes place at year 7 at a distance of nearly 7 light-years, so he won't see it until nearly year 14. Terence sees an Outbound Leg of long duration, and an Inbound Leg of very short duration.
So there's the fundamental asymmetry: the switch from redshift to blueshift occurs at Stella's turnaround. Stella sees Terence's telescopic image age slowly on her Outbound Leg, but the image more than makes up for its dawdling on the Inbound Leg. Terence sees Stella's image off to a slow start too, but here the image's final burst of rapid aging comes too late to win the race.
See the section titled Too Many Analyses for a spacetime diagram of the Doppler Shift Analysis.