Popper and Dingle on special relativity and the issue of symmetry

Karl Popper and Herbert Dingle engaged in a fascinating debate concerning the kind of theory the special theory of relativity is. One of the issues was whether applications of the theory could be made consistent with the principle of relativity, a cornerstone of the theory itself. The principle of relativity seems to imply some sort of symmetry in results obtained for similar experiments as observed in two different inertial reference frames. Peter Hayes has recently dealt with the Dingle–Popper debate on this matter, as well as other issues. The present paper seeks to clarify what kind of symmetry is appropriate in a situation discussed by Popper, Dingle, and Hayes.