If complementing a magic square results in another aspect of the square itself, the square is called self-complement(ary). These include the associative squares and squares in which all complement pairs are side-to-side, i.e., left-to-right or top-to-bottom, symmetric.
All odd order self-complement magic squares are associative.
Doubly-even self-complement magic squares include associative and side-to-side symmetric; singly-even are only side-to-side symmetric, (there are no singly-even associative magic squares).
There are 352 order 4 self-complement magic squares, (48 associative, 304 side-to-side symmetric).
SelfComplementSquares makes side-to-side symmetric squares using the double border method:
Heinz, Harvey "Self-similar Magic Squares" http://recmath.org/Magic%20Squares/Self-similar.htm.
Suzuki, Mutsumi "Magic Squares" http://web.archive.org/web/20060709213003/http://mathforum.org/te/exchange/hosted/suzuki/MagicSquare.html