I have repeated the search using all twelve angles, six at each centroid (each such search takes about 12 hours on my laptop!). I present below the mean of the sines of six angles at one of two rings, with the colour code indicating the centroid-centroid distance.

For examples where the mean sine at each centroid is close to 1.0, the centroid-centroid vector is exactly perpendicular to the plane of both rings. Where the means are < 1.0, the centroid-centroid vector is inclined wrt the ring planes and when the values for both rings are ~equal, it indicates that the two rings are dislocated but co-planar.

Another view of the data is shown below in which the mean of the sine of all twelve angles is plotted against the centroid-centroid distance. Another interesting candidate for inspection emerges, ringed in red (dataDOI: 10.5517/cc13skn3)

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