  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!]
