Some Shifts Don't Alter the Magic Square Sums

 There are some but not all shifts of the contents of a magic square which do not alter the sums. Below the first magic square is shifted one position left in the second square and the first column is moved to the last. The sums for the rows and columns are unaltered but sum for the diagonals and the center and corners have been changed. Repeating the operation produces a square in which all these sums are unchanged from the original. Another way of looking at the result is that the left and right halves can be exchanged. The same is true for vertical shifts since they are equivalent to a diagonal flip followed by a horizontal shift and a second diagonal shift back to the original position.

So we can say that the sums of magic squares are invariant with respect to reflections about an axis and translations including rotations. So from any given magic square we can easily find a number of others.