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Thanks! That's a good one!
JFYI: I've now added this as clmul application to the current xbitmanip draft at https://raw.githubusercontent.com/cliffordwolf/xbitmanip/master/xbitmanip-draft.pdf … (you are now listed as contributor to xbitmanip, btw) and I've added the following invariant to my test cases.pic.twitter.com/zRtU4R5ztX
I've done some exploration of this using SyGuS and quantified SMT. See for example http://svn.clifford.at/handicraft/2018/xbitpermsyn/ …. (Also submitted some SyGuS benchmarks.) A big unsolved problem seems to be generating programs that contain large (word-sized) magic constants.
Since the Gray code is bit-reversed polynomial multiplication by x + 1, whose inverse modulo x^32 is all 1s, you can also have grev32(clmul32(grev32(x ^ (x >> 1), 31), -1), 31) == x.
Oh, yes. That also works. 
However, my "clmul_gray2" uses one instruction fewer. 
So current status of gray code vs clmul invariants:pic.twitter.com/t06o5v9TCW
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