Here is a proof using the indices (in "conventional" notation):
For every pair (i,j) of indices to the nxn matrix G,
G[i,j] = G[j,i] {symmetry of G}
= G[j+k,i+k] {for every k, by rotational symmetry of G}
= G[n-i-1,n-j-1)] {by letting k=n-(i+j+1)}
and the last expression is the mirror image of G[i,j]
(Indexing is origin 0 and the index arithmetic is residue mod n)
/J
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