Operations on Magic Squares that Don't Change the Sums

  There are a number of operations which do not change the relative positions of the numbers in a magic square and so do not alter the sums. These include the operations that project a square onto itself like flips about a line through the the square's center. We can choose either horizontal or vertical flips, diagonal flips or any combination.

The first magic square above is the simple one found earlier. The second one is the result of flipping the first about a vertical line through the center. The third is the first square flipped about the 1-16 diagonal. It doesn't matter what the values since only their position or coordinates are changes. The same is true for a 90° rotation.