  In [4], the continuous cluster category was introduced. This is a topological
category whose space of isomorphism classes of indecomposable objects forms a
Moebius band. It was found in [4] that, in order to have a continuously
triangulated structure on this category, one needs at least two copies of each
indecomposable object forming a 2-fold covering space of the Moebius band. This
paper classifies all continuous triangulations of finite coverings of the basic
continuous cluster category. This includes the connected 2-fold covering of
Igusa-Todorov [4], the disconnected 2-fold covering of Orlov [6] and a third
unexpected continuously add-triangulated 2-fold covering of the Moebius strip
category.
