[math-fun] "Proizvolov's identity"

A recent post on the (excellent) blog "Futility Closet" http://www.futilitycloset.com/2014/08/21/proizvolovs-identity/ cites without proof the following theorem: Take the numbers 1..2n. Divide them into two subsets of size n. Write one in ascending order and the other in descending order. Then the sum of |a_i-b_i| is always n^2. This isn't difficult to prove, but the proof I found not-difficult to find isn't terribly elegant. Is there a neat "bijective" proof? (There are two proofs at http://www.cut-the-knot.org/Curriculum/Games/ProizvolovGame.shtml neither of which seems very elegant to me. Both are better than my proof, which proceeds by induction on n.) -- g