What does the accelerated twin see in the twin paradox?

Let's have Bob travel out for 30 seconds and back for 30 seconds at 4/5 the speed of light while Alice stays at home. They each have clocks that (according to them) tick once per second.

I. When Bob leaves, his clock and Alice's both show time $0$.

II. As Bob travels and looks back through his telescope, he sees Alice's clock tick once every 3 seconds. But he figures that this is partly because the distance between him and Alice is increasing, so the light from each tick takes longer to reach him than the light from the previous tick. After correcting for this, he concludes that Alice's clock is ticking once every 1.67 seconds. That is, her clock is running slow, but not by as much as you'd think if you just measured the interval between arriving ticks and failed to correct for travel times.

III. When Bob's clock reads 30 seconds, he receives the light from Alice's tenth tick (that is, the image in his telescope shows Alice's clock reading 10), and calculates that Alice's clock now reads "18 seconds" (though the light from the most recent 8 ticks hasn't reached him yet).

IV. Now Bob turns around and decides that he and Alice are (and always have been) moving closer together, not farther apart. His telescope still shows Alice's clock reading "10", but now, unlike before, Bob believes that each tick has traveled less distance than the one before it. This causes him to recalculate that Alice's clock currently reads "82 seconds".

V. Bob now travels home, during which time his clock ticks 30 times, and he sees Alice's clock tick 90 times (advancing from 10 to 100). But (according to Bob) 72 of these ticks had already occurred before he turned around, though he had not yet received the light from those ticks. Therefore, according to Bob, the remaining 18 of those 90 ticks take place during his 30-second return journey. So he sees 3 ticks a second, but calculates that Alice's clock is ticking once every 1.67 seconds.

VI. Bob arrives home. His clock reads 60 seconds. Alice's clock reads 100 seconds.

Where did I get all this? I drew the spacetime diagram. So should you.

Added in response to the followup comment by @Razor:

1) Say, for convenience, the waves/photons emitted by Alice are seperated by 1 second in Alice’s frame and Bob continuously receives them. So, at the moment Bob turns, does the time interval collapse?

I can't answer this with confidence because I haven't the foggiest idea what it would mean for a time interval to "collapse". But my best guess is that whatever you mean by this, the answer is no.

2) Say, for convenience, the waves/photons emitted by Alice are seperated by 1 second in Alice’s frame and Bob continuously receives them....Surely he cannot miss any photons (since photon absorption by Bob is something absolute). He absorbs all the photons or did he just miss Alice aging?

Obviously, Bob's turning cannot affect the paths of the photons. So: Suppose that just as he turns, he receives a photon that Alice sent when she was exactly 20 years old.

Before the turn, Bob says: Here comes a photon that Alice sent when she was 20. Right now she's pretty far away, but when she was 20, she was a lot closer. So the photon has taken a pretty short time to get here, which means that right now she's a little older than 20.

After the turn, Bob says: Here comes a photon that Alice sent when she was 20. Right now she's pretty far away, but when she was 20, she was even farther. So the photon has taken a really long time to get here, which means that right now she's a lot older than 20.

Why does Bob say these things? Because before the turn, he's employing a frame in which Alice is moving away from him (and always has been), whereas after the turn, he's employing a frame in which Alice is moving toward him (and always has been. In the first frame, she used to be closer than she is now; in the second, she used to be farther than she is now.

3) As I said eighteen months ago when I first answered this question: If you want to discover these things for yourself, start by drawing the spacetime diagram.

I made some videos about this a few years back, on YouTube. It shows what the traveler sees on a variety of clocks in the scene, for the entire duration of the journey.

I would suggest "Earth gravity front view" to start with ("Earth gravity" represents the value of acceleration in this context!)

The terse explanatory text is reproduced here:

The twin "paradox" (in quotes because it is NOT a real paradox!) is a basic teaching scenario for Special Relativity: http://math.ucr.edu/home/baez/physics...

This is a series of visualizations of the journey from the point of view of the traveling twin, who flies 20 light years away from his home station then returns. The journey consists of four parts, joined together. The first quarter is an acceleration away from the station. During the second and third quarters the ship accelerates towards the station (so that at the half way point the ship is stationary 20 light years away). In the fourth quarter the ship accelerates away from the station in order to come to rest there.

The total coordinate travel time (shown as a red dot in the top left HUD clock) is 43.711/58.918 years for acceleration at earth/moon gravity levels, whilst proper time (green dot) is 12.101/38.694 years. The yellow dot represents the time the traveler would see on the station clock face through a very powerful telescope! The octahedral stations (spaced one light year apart and one light year to the left of the flight path) are all synchronized to coordinate time and rotate once over the course of the whole journey. There is a 2x2 light year wall one light year beyond the far end of the journey, and large rectangular frames every 5 light years. Where a floor is shown it is 1 light year per stripe, and there are small 1 ly milestone spheres along the way, with a larger one every five light years.

The flights are rendered without relativistic effects and then for two values of acceleration (currently earth gravity and moon gravity). The distortion artifacts are due to aberration of light, the Doppler effect and the headlight effect. The earth gravity videos exhibit some nice penrose-terrel "rotation" effects. Magenta markers in a circle show where you really are in the scene (in the sense that things outside the circle are behind you, whilst those inside the circle are in front of you. Doppler shift = gamma for this circle), and grey markers show the circle where the doppler shift is 1 (gamma is in theory directly observable on this circle).