Nope I was too lazy to update the cases of nonparallel "sections"¹, only mentioned it in reply to hdb (whose solution makes mine obsolete).
But I think that hdb's solution resp. your generalization already describe the complete solution room and that all possible magic square can be found there by mapping rows to diagonals.
So since the other way round works you will always find a backprojection from magic to "limbic" square.
Too tired to dig into proving it, but looking into the literature for magic squares should show it's trivial.
(unproven opinion)
Cheers Rolf
( addicted to the Perl Programming Language)
update
¹) and I'm still not sure if L~R really wants them.
