I recently bought a copy of Einstein's booklet Relativity, in which Einstein attempts to describe the basics of special relativity to laypersons. In Chapter 12, Einstein explains how the Lorentz transformations can be used to compute how the relative motion of frames of reference affects measuring rods and clocks.
Suppose I stand on a railway boarding platform, and my friend stands on a railway car which is moving past the platform. I have one frame of reference, and my friend has a second frame of reference which is in motion relative to the first. Each of us holds a meter stick and a stopwatch. At the moment he passes me, we both hold out our meter sticks and stopwatches to be compared. According to the Lorentz transforms, in my frame of reference I will observe his meter stick to be shorter than mine, and I will observe his watch to be ticking slower than mine. Similarly, in his frame of reference he will observe my meter stick to be shorter than his, and he will observe my stopwatch to be ticking slower than his own.
It seems odd to me that we would both observe our meter sticks to be shorter than the other person's, but I can accept this. What I can't reconcile is the stopwatches. Suppose my friend remains in motion for some time, but then his train loops back around to the station, stops, and we compare stopwatches. We now have the same frame of reference. Someone's watch must be ahead of the other person's, but whose?
Let me modify the experiment a bit. I have two stopwatches of identical construction which are synchronized. I set watch A on a table, and I put watch B on the edge of a merry-go-round and spin it very fast for a long time. Here the frames of reference are the table (k) and a point on the edge of the merry-go-round (k'). Because k' is moving relative to k, from perspective k B will tick slower. However, k is also moving with respect to k', so from perspective k' A is ticking slower. When I stop spinning the merry-go-round, which watch will be behind?