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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
Neat, the high part does the inversion itself. You can also compute the xor of any number of rotations of a word; for example SHA-256's S1 is doable as clmul(e, 0x4200080) ^ clmulh(e, 0x4200080).
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