  Many extended conformal algebras with one generator in addition to the
Virasoro field as well as Casimir algebras have non-trivial outer automorphisms
which enables one to impose `twisted' boundary conditions on the chiral fields.
We study their effect on the highest weight representations. We give formulae
for the enlarged rational conformal field theories in both series of W-algebras
with two generators and conjecture a general formula for the additional models
in the minimal series of Casimir algebras. A third series of W-algebras with
two generators which includes the spin three algebra at $c=-2$ also has
finitely many additional fields in the twisted sector although the model itself
is 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.
