a polynomial representation of boolean functions
Well done on your independent discovery.

This is a Hadamard matrix of Sylvester type, written in a different order. There is a massive literature on this topic. It is used in coding theory, cryptography, is directly related to Reed-Muller codes of degree 1, it can be used to obtain best affine approximations of functions, etc.

Ryan O'Donnell's notes on Analysis of Boolean functions available here which are now published as a book, Claude Carlet's chapters on Boolean 
functions here are easily accessible online.

Just use the order $[1,X,Y,XY,Z,ZX,ZY,ZXY]$ which corresponds to lexicographic order over the integers, and apply the corresponding permutation to the columns as well. 

In that order your matrix is simply the 3-fold Kronecker product $$H_2 \otimes H_2\otimes H_2$$of the basic 2 by 2 Hadamard matrix.

$$
H_2=\left(\begin{array}{cc} 1 &amp; 1 \\ 1 &amp; -1 \end{array}
\right),
$$

This paper considers efficient evaluation of Hadamard representation coefficients.