Acceleration or Deacceleration. Can either be identified? An accelerated clock will measure a smaller elapsed time between two events than that measured by a non-accelerated (inertial) clock between the same two events.
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Experiment #1
Imagine that you have a space station. Docked at the space station you have two rockets.
Each rocket is pointing away from the space station. One is on the left side(1) and one is on the right(2).
At 12:00 noon both rockets depart and quickly accelerated in opposite directions to 260,000 kilometers per second. Once reaching this speed they both send a radio pulse signal back to the space station and do the same an hour later. At this speed their clocks are ticking at half the speed of which the clocks are ticking back at the space station. 
Thus at the space station the time measured between the two radio pulse signals sent from the rocket off to the left would be two hours rather than one. Also at the space station the time measured between the two radio pulse signals sent from the rocket off to the right would also be two hours rather than just one.
Is this not true without question?
Please answer this question prior to proceeding onward to the following paragraphs.
Experiment #2
However, what if during sleep hours, the space station along with the two rockets, were all accelerated off to the right to a velocity of 260,000 kilometers per second, and done so by let's say Aliens from far away. 
Those within the space station and in the rockets, due to being asleep, were therefore completely unaware of this "Acceleration".
If the above two 12:00 noon rocket departures were repeated under these new conditions, will those at the space station obtain the same measurement results? After all, the rocket on the left will deaccelerate back to the original state, yet the rocket on the right will have been accelerated a second time.
( Please note insightful correction made by Godparticle down below. )
 A: In the first case, the time interval between two radio signals is not exactly two hours, according to the clock in the station. The radio signal sent at the time when the rocket was accelerated to $26\times10^4kms^{-1}$ has to travel certain distance to the station which costs time. Radio pulse sent later an hour (according to space ship clock) needs to travel longer distance than the first radio signal. So, the time is going to be greater than two hours according to the clock in the station..
A: Three comments. 


*

*Your claim about the arrival time of the dilated clock signals in experiment 1 ignores the light speed retardation of the signals (which is larger at each tick), but this does not affect the basic outcome. 

*There is no absolute rest frame so the two cases are identical and will yield identical observations by the scientists on-board the space station. 
Judging from the wording of your title, I would guess that what is causing the confusion is your implicit assumption that there exists a unique condition called "stopped". All inertial frames are equally valid.

*You may also postulate the existence of an observer who remains in the rest frame of experiment 1 and watches experiment 2 (perhaps this is the alien who played the joke). From his point of view the dilation effect is reversed between the space station and the left-bound rocket. This is just another way of stating the starting point of the basic twin-pardox.
It is true that the station sees the rocket's clocks as running slow and that the rockets (both of them) see the station's clock as running slow (they also see the clock on their opposite number running slower still). These facts may seem to be contradictory, but they are not because space-time does not work the way you think it does.
See the fabulous image that Alfred Centauri dug up and posted to another answer  to (maybe, sorta) get an intuitive handle on this.
A: Sean, I'm not quite sure if I understood you correctly in everything, however ...
We are running into similar problem here that we have in a classical SR situation. Let's say that the two rockets are moving in opposite directions, but at constant $v$ (we don't care at the moment how they developed their speeds) with respect to the station. If we assume the station is stationary, and because the direction of movement (right, left, up, down ...) does not matter for the calculation of time dilatation, as there is no "preferred direction", we must also assume - since both rockets are moving at $v$ wrt. to the station - that clocks on both rockets should be dilated by exactly the same amount of $\Delta t'$ as compared to the station's clock.
However, if we change frames and re-run our reasoning from the POV of any of the rockets, we can see that none of the three clocks should be showing the same time dilatation. The station is moving at $v$ wrt. to the right rocket, and the left rocket is moving at $2v$ wrt. to the right rocket. 
So what would the clocks actually show if we conducted real-life experiment? Well, I only wanted to show the problem. [I'm not going to answer, because this is not what you asked about, and what I might say would likely get me heavily downvoted, which is not what I am asking for.]

Back to accelerations. How do we define acceleration/deceleration in kinematics? Acceleration means (continuous) increase of the velocity of one object relative to another, and deceleration means decrease of the velocity of one object relative to another. However, we should note that in space both actions require turning the engines on (otherwise none will happen), and so they are locally indistinguishable. So, both acceleration and deceleration require that the object undergoes proper acceleration, and are therefore relative.
On the other hand, GR shows time dilatation due to acceleration despite the fact that the distance between two clocks (and therefore relative speed) is constant [unless we take Einstein's equivalence principle literally and assume that Earth is physically accelerating (and therefore expanding)]. For this reason, we cannot assume that the direction of the change of velocity (whether the relative speed between two objects is increasing or decreasing, i.e. whether we are dealing with relative acceleration or deceleration) influences the direction of time dilatation. 
Still not so obvious? OK, back to the accelerating rockets. From the POV of the station both rockets are just accelerating, and the direction of movement does not matter (sending a rocket in any direction should produce exactly the same effect). Therefore they both show exactly the same time dilatation. After some time they turn off their engines and continue their travel at constant $v$, so there is only classical SR dilatation at play now (which is not our concern here). Then the station changes its (relative) movement in such a way that the difference in velocity between the station and the right rocket becomes increasingly smaller. Therefore it is decreasing it's relative velocity wrt. to this rocket, and so we can say it is decelerating (if I understood you correctly, it continues until it becomes stationary wrt. to the right rocket, correct?). However, when it is decelerating wrt. to the right rocket, it is also accelerating wrt. to the left rocket: the difference in speed between them is increasing from $v$ toward $2v$. 
So what's the final conclusion then? Well, if we assumed that the initial continuous change of velocity of both rockets produced certain effect (slowing their clocks down by exactly the same value wrt. the station) regardless of the direction of their movement, than we must also assume that the continuous change of the station's velocity must produce exactly the same effect. (We can always leave a single clock drifting in space right where the station was located, and the relation between this drifting clock and the clock on the station must necessarily mirror that between the station and the rockets before.) So it does not matter if one frame is decreasing or increasing its velocity wrt. to some other frame. What matters is only the fact that its velocity is changing.
Therefore, as far as time dilatation is concerned, we may simply assume that acceleration is just acceleration. (Revealing, isn't it? :) ) 
Which means that unless experiments show otherwise (which would be quite surprising), we must assume that what really counts is only the so-called proper acceleration, i.e. the indication of an accelerometer located on board of each of the three objects, and not their respective velocity (change). From the perspective of an object changing its velocity, the direction of this change (wrt. to some external object) does not matter. Whether we call it acceleration or deceleration makes no difference, because its essentially the same thing.
Or, put differently, since - according to SR - we don't know if a given inertial frame is moving or not, than we also don't know, if, when it changes its velocity, it is actually accelerating or decelerating. It simply undergoes proper acceleration in some direction. All the rest is relative and therefore irrelevant.
Is this what you were asking about?
