  We establish explicit duality transformations for systems of M q-state Potts
models coupled through their local energy density, generalising known results
for M=1,2,3. The M-dimensional space of coupling constants contains a selfdual
sub-manifold of dimension D_M = [M/2].
  For the case M=4, the variation of the effective central charge along the
selfdual surface is investigated by numerical transfer matrix techniques.
Evidence is given for the existence of a family of critical points,
corresponding to conformal field theories with an extended S_M symmetry
algebra.
