  Using the example of Liouville theory, we show how the separation into left-
and rightmoving degrees of freedom of a nonrational conformal field theory can
be made explicit in terms of its integrable structure. The key observation is
that there exist separate Baxter Q-operators for left- and right-moving degrees
of freedom. Combining a study of the analytic properties of the Q-operators
with Sklyanin's Separation of Variables Method leads to a complete
characterization of the spectrum. Taking the continuum limit allows us in
particular to rederive the Liouville reflection amplitude using only the
integrable structure.
