It may help you quite a bit to realize that some linear algebra shows that all solutions are of the form:

  x+y    x-z  x-y+z
 x-2y+z   x  x+2y-z
 x+y-z   x+z  x-y
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By rotating and reflecting we can make the largest corner be x+y, and we can insist that x-z > x-2y+z. In this case we have 0 < z < y The condition that all values be in the range 1..26 is satisfied if 1 <= x-2y+z < x+2y-z <= 26. Uniqueness is satisfied if 2z != y.

We can actually make a stronger statement. If 2z < y, then the elements fall in the order x-2y+z, x-y, x-y+z, x-z, x, x+z, x+y-z, x+y, x+2y-z and if y < 2z then the elements fall in the order x-2y+z, x-y, x-z, x-y+z, x, x+y-z, x+z x+y, x+2y-z.

With this many conditions, it should not be hard to enumerate the magic squares up to symmetry. And with some cleverness, I believe you don't even have to enumerate them all.

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