Regge and Okamoto symmetries
  We will relate the surprising Regge symmetry of the Racah-Wigner 6j symbols to the surprising Okamoto symmetry of the Painleve VI differential equation. This then presents the opportunity to give a conceptual derivation of the Regge symmetry, as the representation theoretic analogue of the author's previous derivation of the Okamoto symmetry.   [The resulting derivation is quite simple, so it would be surprising if it has not been previously observed. Any references would be appreciated!] 
Philip Boalch