3^2-2^2 =!= 2^2-3^2.
(You can exchange a and b in, say a^2+b^2, because 2^2+3^2=3^2+2^2)
(You can exchange a and b in, say a^2+b^2, because 2^2+3^2=3^2+2^2)
Your rewriting is of course true for all a,b and might be used in an algebraic proof. But this transformation is not at all shown in the geometric proof.
That's not what anyone is saying.
The point is that a+b is symmetric in a <-> b and a-b is anti-symmetric. Both left and right side are anti-symmetric.