There is a similar DSP trick here, but I don't remember the details exactly.
I read about it somewhere, some while ago. It has to do with figuring out fabric pattern matches regardless of the orientation. So you may want to research on that.
Grab a circle sample. Do sums along spokes of the circle to get a circumference profile. Then they did a DFT on that (it is inherently circular after all). Toss the phase information (make it orientation independent) and make a comparison.
Then they could tell whether two fabrics had the same pattern.
Your problem is similar.
It seems to me, without trying it first, that the characteristics of the pre DFT profile should reveal the orientation. Doing standard deviations along the spokes instead of sums should work better, maybe both.
Now, if you had an oriented reference image, you could use their technique.
Ced
Your precision requirements are rather strict.
I gave this a whack. Taking the sum of the absolute values of the differences between two subsequent points along the spoke for each color.
Here is a graph of around the circumference. Your value is plotted with the white markers.
You can sort of see it, but I don't think this is going to work for you. Sorry.