Why isn't time contraction a thing? I'm very naive about relativity, but if it could be possible that Alice ages less than Bob in a journey, doesn't it logically follow that from the point of view of Alice, Bob has, at some point, gone through time contraction? 
Additionally, Wikipedia provides a specific example, and by applying the Doppler shift concludes :

during the trip back, both twins see their sibling's clock going 3 times faster than their own.

Isn't this the exact opposite of time dilation? If not, how different is time dilation from the observation of slowed-down clocks?
 A: A proper exposition of the effects implied by special relativity concentrates on the following: when your motion does not involve any acceleration then the amount of time that elapses for you is a maximum.
So: we have Bob not accelerating. Let's say he is moving in the sense that he is moving forward in time. Alice is on a trajectory that involves an acceleration phase away from Bob, and a U-turn in any way, and then rejoining Bob. At the instant of rejoining they are once again right next to each other, so they can make a direct comparison of how much proper time has elapsed for each of them.
On the other hand, there is ambiguity when they compare their clocks while Alice is far away from Bob, because then you have to take transmission delay into account. Likewise, comparing clocks while Alice is on the move is ambiguous, because then you have to take classical Doppler effect into account.
The effect that is specific to special relativity is laid bare when you remove all other effects. You set up a scenario where at the moments of comparison there is no transmission delay, and no Doppler effect.
(So anytime you encounter an exposition of special relativity where a scenario is offered that includes transmission delay and/or classical Doppler effect you can just stop reading that exposition; it won't teach you.)
So you focus on the scenario of Bob not accelerating, just moving forward in time, and Alice accelerating away, making a U-turn, and rejoining.
For Alice, by traveling more spatial distance than Bob, less proper time elapses. That is: there is something you can actively do so that for you less proper time elapses.
On the other hand, there is nothing that Bob can actively do so that for him even more proper time elapses; he is already moving forward in time along the spatially shortest path, so he is already maxing out the amount of proper time elapsing for him.
So it's not a symmetrical situation; Bob is already maxing out, Alice can make the amount of proper time that elapses for her smaller by increasing the spatial distance she travels (in the same amount of Bob's proper time.)




[The following addition, two years after this answer was originally submitted, is in response to a comment]

We have that special relativity describes the nature of spacetime with the Minkowski metric.
As we know, Euclidean space has a spatial invariant: Pythagorean length:
$$ l^2 = x^2 + y^2 + z^2 \tag{1} $$
Minkowski spacetime has the invariant spacetime interval.
As is customary when discussing fundamenals: in the following expression the speed of light is used as unit of speed. That is, $c$ is set to 1
The expression for the invariant spacetime interval:
$$ t^2 - x^2 - y^2 - z^2 \tag{2} $$
When the coordinate system is arranged to have the x-axis parallel to the direction of motion it is sufficient to specify the x-coordinate only:
The expression for the invariant spacetime interval:
$$ t^2 - x^2  \tag{3}  $$
As we know: if you have two inertial coordinate systems, 'a' and 'b', that have a uniform velocity relative to each other, then the invariance of the spacetime interval implies the following relation:
$$ t_a^2 - x_a^2  =  t_b^2 - x_b^2  \tag{4}  $$
The Lorentz factor is another way of expressing the Minkowski metric.
(In general we have in physics: derivations can often be run in both directions )

The comment by GumbyTheGreen gives the impresssion that GumbyTheGreen believes that the Lorentz factor is a standalone concept, unrelated to the Minkowski metric.
In physics the concepts 'distance traveled' and 'velocity' are interconnected, because velocity is the derivative of position coordinate with respect to the time coordinate. It is not clear whether GumbyTheGreen is aware of this.

We have in the physics of special relativity that we have limited knowledge of causal structure of Minkowski spacetime. The Minkowski metric specifies a relation; in and of itself the Minkowski metric does not attribute causality.
In my 2019 answer I made sure I was not attributing causality.
I my answer I kept the wording neutral; if Alice acts in such-and-such a way the end result will be so-and-so.
Of course: there must be a causality somewhere, but I make sure that my wording does not single out some specific supposition of causality (as that would be arbitrary).

It appears singling out some arbitrary supposition of causality is exactly what GumbyTheGreen is insisting on. GumbyTheGreen is insisting: difference in elapsed proper time is not related to the invariance of the spacetime interval, but it is a function of the Lorentz gamma factor.

Now: let's take up the invariant spacetime interval again.
We describe the physics taking place using a coordinate system that is co-moving with Bob, who remains in inertial motion the entire time.
Using that inertial coordinate system: (4) reduces to this relation:
$$ t_a^2 - x_a^2  =  t_b^2  \tag{5}  $$
Of course, the journey of Alice includes phases of acceleration. In particular, in order to rejoin Bob Alice has to make a U-turn.
The way to accommodate that is to divide the journey of Alice into sections. (5) is not only valid for extended intervals of distance; (5) is valid on any length of interval of distance, down to infinitisimally short intervals of distance. Also, we can concatenate any number of sections into an entire journey, and tally the result.
Generally:
Various calculation strategies can be employed. The calculation can be set up to use the invariance of the spacetime interval, or it can be set up to use Lorentz gamma factor; the end result should be the same.

 
Addressing GumbyTheGreen personally:  
If you disagree with the above then I propose that you submit that disagreement to the stackexchange *community*. That is: submit it as a *new question*.
For instance, you could phrase it as follows:
"In a recent answer a stackexchange contributor stated that the invariance of the spacetime interval and the Lorentz gamma factor are related to each other. Is that correct?"




I need to address an ambiguity here.
This ambiguity is not present in a spacetime diagram, since that is a geometric representation, and my thinking process in the form of visualization (spacetime diagram). I think that is why I previously didn't spot the ambiguity.
For the case of 2D euclidean space, represented with cartesian coordinates.
Let A be one coordinate system, and B be another coordinate system, rotated with respect to A. Let the rotation be such that a vector of 5 units long that is along the x-axis of B comes out as having components of 4 units and 3 units along the axes of coordinate system A.
The case of journeys in spacetime
Let me switch to a new set of names for the protagonists of the thought experiment, as I want to use three protagonists now. I will refer to them as Terence, Stella, and Galla.
Terence remains in inertial motion the entire time. I will refer to the inertial coordinate system that is co-moving with Terence as 'the global inertial coordinate system'.
Stella and Galla separately go on a journey, away and then back again to Terence. At time coordinate t_1 (Terence time) Stella and Galla set out on their respective journeys. Galla's journey is planned to take her further away from Terence  than Stella will travel. The journeys are plotted such that Stella and Galla will both rejoin Terence at time coordinate t_2 (Terence time).
In this evaluation the distance values used are the distance values as they are plotted in the global inertial coordinate system.
The intended evaluation - but I didn't state that explicitly - is that the position coordinates, velocity values, and acceleration values of the motion with respect to the global inertial coordinate system are to be used.
Then, in a comment, I used the phrasing: "when zigzagging you accumulate a lot of acceleration"
In retrospect I should have phrased that as follows:
"When the traveller is zigzagging the traveller accumulates a lot of acceleration relative to the inertial coordinate system."
All statements about motion are to be read in terms of motion with respect to the global inertial coordinate system.
Galla travels further away from Terence than Stella does, and in order to do that Galla must travel faster than Stella does. Again, this statement about velocity is to be read in terms of the velocity of Stella/Galla with respect to the global inertial coordinate system.
In this scenario the motion of Galla with respect to the global inertial coordinate system covers more spatial distance than the motion of Stella with respect to the global inertial coordinate system.
A: Time dilation means that one of the observers experiences a longer time period than another, as a consequence of a relativistic velocity (comparable to the speed of light c).
Time contraction would mean that traveling close to the speed "c" would make the time interval for the persion traveling fast to be smaller than for the one standing still, thus having him "travel backwards in time".
Best way to see why it is NOT a thing is to see the formula for time dilation:
$$t'=\frac{t}{\sqrt{1-\frac{v^2}{c^2}}}$$
as you see, the equation only allows for t', which is the time that a observer that is NOT moving, to be greater than the time that the person moving experiences.
A: The question posed confuses three separate issues- namely time-dilation, the twin paradox and the Doppler effect- which need to be disentangled and addressed separately.
The Doppler effect, firstly, governs how the timing of events is seen at a distance when they are moving relative to an observer. It is a result of the fact that owing to their relative motion the observer is not at a fixed distance from the events. If you look at a distant clock while travelling away from it, light from each successive tick has farther to travel, and so takes longer to reach you, so the ticks appear slowed down. Conversely, if you are moving towards a clock its ticks will appear to be speeded up.
The so called twin paradox (which is not a paradox) relates to the fact that an observer who moves inertially between two events will record a longer elapsed time than someone whose path between the two events involves acceleration. In the classic version one twin stays on Earth while the other zooms away, turns around and comes back, being younger than the stay-at-home twin upon their return. You would be quite entitled to say that the  travelling twin has aged less than the stay-at-home twin, or that the stay-at-home twin has aged more than the travelling twin- the two statements are logically equivalent.
Time dilation in SR refers to the fact that when a clock moves inertially between two stationary points, the elapsed time for the clock is always less than the difference between the coordinate times at the two points in their rest frame. It is a consequence of the relativity of simultaneity. If you were coasting inertially through some frame and compared the time on your watch with that shown on the clocks you were passing, you would see that your watch was falling gradually further and further behind the time on each clock you pass. You would be wrong to think your watch was running slow compared to the clocks, or that they were ticking faster than your watch- instead what is happening is that the clocks are all systematically out of synch in your frame of reference, each clock you encounter being further ahead in time than the one before it.
