Can someone prove that if $S$ is a filled magic square, and $T$ is obtained from $S$ by switching two rows or two columns, then $T$ is also a filled magic square.
So an example that I came up with was: 
and 
are magic squares related by the exchange of the first two columns.
Suppose we switch two columns. Clearly, the column sums are unchanged by this operation. If they were all equal before the switch, they are still equal because the columns themselves are exactly the same.
In each row, the column switch operation leaves all the numbers the same; all it does is reorder the numbers by switching positions of two of the numbers in the row. Since addition is commutative, the row sum is unchanged.
The same argument shows that switching rows does not change the row or column sums.
However, the argument does not show that the diagonal sums are unchanged, and it cannot, because your example shows that they can change.