In the definition of Virasoro algebra, there is a following condition on the generators:
$[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n,0}$
Now, Regge symmetry is the following statement:
...a generic Euclidean tetrahedron gives rise to a family of twelve scissors-congruent but non-congruent tetrahedra
This is nicely explained in Roberts' paper "Classical 6j-symbols and the tetrahedron," http://arxiv.org/abs/math-ph/9812013 . Basically, this follows from an asymptotic formula for the (classical) $6j$-symbol.
Now, my question is the following: is this the same 12 in both cases? I was told that there is a restatement of Regge symmetry that has to do with Virasoro algebra. Could someone elaborate on this connection?