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Rotational symmetry is a nice play around thingy. Simply said, an approach that you could use is template based matching. There are some general steps to take in order to perform this.

  1. Rotate a given image (imgBase) for some degrees and call it imgRotated
  2. For the first image, define some interesting points by performing an interesting feature point detector like SURF, SIFT, ORB, ...
  3. Use a set of features defined on the imgBase as locations for test. Feed regions of interest around those points as templates to a descriptor and create a representation for it. Many aproaches are here possible, look at this link :
  4. Apply same descriptor at same locations in imgRotated
  5. Match the descriptors for each point and define the amount of equality which will be an amount of rotational symmetry.

I do not have a solution, but given this comments, I think you got some starting points right.