  We extend Poincar\'e's theory of orientation-preserving homeomorphisms from
the circle to circloids with decomposable boundary. As special cases, this
includes both decomposable cofrontiers and decomposable cobasin boundaries.
More precisely, we show that if the rotation number on an invariant circloid
$A$ of a surface homeomorphism is irrational and the boundary of $A$ is
decomposable, then the dynamics are monotonically semiconjugate to the
respective irrational rotation. This complements classical results by Barge and
Gillette on the equivalence between rational rotation numbers and the existence
of periodic orbits and yields a direct analogue to the Poincar\'e
Classification Theorem for circle homeomorphisms. Moreover, we show that the
semiconjugacy can be obtained as the composition of a monotone circle map with
a `universal factor map', only depending on the topological structure of the
circloid. This implies, in particular, that the monotone semiconjugacy is
unique up to post-composition with a rotation. If, in addition, $A$ is a
minimal set, then the semiconjugacy is almost one-to-one if and only if there
exists a biaccessible point. In this case, the dynamics on $A$ are almost
automorphic. Conversely, we use the Anosov-Katok method to build a
$C^\infty$-example where all fibres of the semiconjugacy are non-trivial.
