IMO GP is about the labelling: Instead of a2-b2 use longer2-shorter2 (and on visual representation one side will always be longer, as per GPs explanation).

Diagram doesn't show proof for "shorter2-longer2". I believe showing negative area would spark more controversy (imagine that negative area is painted orange, visual area would still be positive).

Shorter2-longer2 gives you same absolute value with reverse signed, so it feels symmetric to me (can't remember what the formal definition is), i.e:

    1.   a2 -  b2  =  ( a-b)(a+b)  # * -1
    2. -(-a2 + b2) = -(-a+b)(a+b)
    3. -(b2 -  a2) = -( b-a)(b+a)  # / -1
    4.   b2 -  a2  =  ( b-a)(b+a)

Edit: https://imgur.com/a/olETWfr - crude image with labels swap as it seems it's bringing some controversy.