special relativity - Is the time dilation experiment for real?

A popular experiment is described in books on special relativity, that claims that an atomic clock was flown at high speed and later compared with a similar atomic clock on earth, and it was found that the atomic clock that was flown at high speed was slower than the one on earth. Was such an experiment really performed? When, where and by who? Is the mentioned result the real and correct result of the experiment?

The reason I think this is absolutely impossible is that when the clock was flown on a plane, the clock on the plane saw that the clock on earth was moving at high speed relative to it, and so clock on the plane saw that the clock on earth was going slow, therefore later when the two clocks met up, it should have been the earth clock which should have been slower than the plane clock, and not vice versa.

Because of this, it would be absolutely impossible to tell which of the two clocks would be slower than the other at the end of the journey. Each clock would see the other clock as slower AND themselves as faster and therefore they would not agree on their observations. So if the experiment was really performed, then it could NOT definitely be the plane clock that was slower.

So how do you explain this anomaly, and if the experiment was really performed, then why did it yield the claimed result and not the opposite?

I agree that there is a difficulty in this experiment ie. the plane clock went around the earth in a circle and therefore had acceleration. But the plane clock saw that the earth clock also went in a circle in the opposite direction (assuming that the plane flew at ground level). But please don't try to tell me that the plane clock was slower due to the acceleration in the circular path, because special relativity says that time dilation is due to relative velocity and not due to acceleration. [If you do say this then please explain why the time dilation of the plane clock was due to its acceleration]

When the two clocks were back together again at the same place, is it possible for the earth people to observe this: (plane clock 12:00:00 earth clock 12:00:01) and the pilot who was flying with the plane clock and came back with it to observe this: (plane clock 12:00:01 earth clock 12:00:00), both observations at the same time in the same place and both contradicting each other?