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File:Walsh permutation 1 3 5 15 17 51 85 255.svg

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Permutation matrix of the Walsh permutation wp( Sloane's A001317(0...7) ) = wp(1,3,5,15,17,51,85,255)
The permutation is row 3 of the triangle Sloane's A197819. It is self-inverse, so the matrix is symmetric.

The odd entries (on the odd places) are the complements of the even entries on their left.
The subsequence of even entries divided by 2 gives wp( (Sloane's A001317(1...7)-1)/2 ) = wp(1,2,7,8,25,42,127):

The permutation is self-inverse, so the matrix is symmetric. Above that the matrix is also symmetric to the antidiagonal.
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