Time dilation is always mutual - both clocks are travelling relative to each other and both must observe that the other clock is slower than themselves. Since motion is relative, and it is not possible to determine which one is noving and which one is still, so it is not possible for just one clock to be slower than the other.
Therefore, in the famous experiment, when the atomic clock was flown around the world, was it observed and recorded that (a) the person travelling with the flying clock found that the clock on earth was slower than the flying clock, whereas (b) the person on earth found that the flying clock was slower than the clock on earth?
In particular, I am interested in the first observation, i.e. for the person travelling with the clock the earth clock was really slower than the flying clock, as that is not the result that one usually hears for this experiment.
For this question, let us consider only the special relativistic part of the time difference, and ignore the general relativistic part, although the actual measured time difference was the sum of both.
For example, https://en.wikipedia.org/wiki/Hafele%E2%80%93Keating_experiment gives results only of time gained by the flying clock. It gives no results for time gained by the lab clock, which makes me wonder whether the time dilation was mutual or not.