Both 8×16 matrices contain the same set of columns.

The lower matrix corresponds to the sorted sequence [2, 28, 42, 52, 74, 84, 98, 124, 130, 156, 170, 180, 202, 212, 226, 252].
The upper matrix corresponds to [170, 180, 130, 156, 226, 252, 202, 212, 42, 52, 2, 28, 98, 124, 74, 84].

Their relationship is described by the self-inverse permutation [10, 11, 8, 9, 14, 15, 12, 13, 2, 3, 0, 1, 6, 7, 4, 5],
which corresponds to the signed permutation [0, ~1, 2, ~3].

Both are related to the noble sequence 0, 30, 40, 54, 72, 86, 96, 126, 128, 158, 168, 182, 200, 214, 224, 254.
They can be derived from it by XORing with 2 or 170 respectively.

Corresponding upper and lower columns are Zhegalkin twins.
(It makes sense to think of the upper columns as Zhegalkin indices, and of the lower columns as truth tables.)
Their XOR is always 00010101, which is little-endian binary for 168.

Corresponding upper and lower rows are equal or complementary.
They are all linear functions, whose Walsh indices are expressed in the small matrices on the left.