Self-complement magic squares

The following square of order 4 is magic with magic sum S=34.

121516
121435
137104
81169

Now each number z is replaced by its complement

selbstkomplementär

which results in a new magic square.

121516
121435
137104
81169
161521
531412
410713
96118

Under certain conditions the resulting magic square will be a reflected or rotated copy of the original magic square. These squares are called self-complement or self-similar magic squares.