Many extended conformal algebras with one generator in addition to the
Virasoro field as well as Casimir algebras have non-trivial outer aut
omorphisms which enables one to impose ''twisted'' boundary conditions
on the chiral fields. We study their effect on the highest-weight rep
resentations. We give formulae for the enlarged rational conformal fie
ld theories in both series of W-algebras with two generators and conje
cture a general formula for the additional models in the minimal serie
s of Casimir algebras. A third series of W-algebras with two generator
s which includes the spin-three algebra at c = -2 also has finitely ma
ny additional fields in the twisted sector although the model itself i
s apparently not rational. The additional fields in the twisted sector
have applications in statistical mechanics as we demonstrate for Z(n)
-quantum spin chains with a particular type of boundary conditions.