Frénicle standard form

A magic square is in Frénicle standard form, named for Bernard Frénicle de Bessy, if the following two conditions apply:
# the element at position [1,1] (top left corner) is the smallest of the four corner elements; and
# the element at position [1,2] (top edge, second from left) is smaller than the element in [2,1] .

This standard form was devised since a magic square remains "essentially similar" if it is rotated or transposed, or flipped so that the order of rows is reversed — there exists 8 different magic squares sharing one standard form. For example, the following magic squares are all essentially similar, with only the final square being in Frénicle standard form:

8 1 6 8 3 4 4 9 2 4 3 8 6 7 2 6 1 8 2 9 4 2 7 6 3 5 7 1 5 9 3 5 7 9 5 1 1 5 9 7 5 3 7 5 3 9 5 1 4 9 2 6 7 2 8 1 6 2 7 6 8 3 4 2 9 4 6 1 8 4 3 8