  It is known that the $k$-faces of the permutohedron $\Pi_n$ are labeled by
(all possible) linearly ordered partitions of the set $[n]=\{1,...,n\}$ into
$(n-k)$ non-empty parts. The incidence relation corresponds to the refinement:
a face $F$ contains a face $F'$ whenever the label of $F'$ refines the label of
$F$. In the paper we consider the cell complex ${CP}$ defined in analogous way,
replacing linear ordering by cyclic ordering. Namely, $k$-cells of the complex
${CP}$ are labeled by (all possible) cyclically ordered partitions of the set
$[n+1]=\{1,...,n, n+1\}$ into $(n+1-k)$ non-empty parts, where $(n+1-k)>2$. The
incidence relation again corresponds to the refinement: a cell $F$ contains a
cell $F'$ whenever the label of $F'$ refines the label of $F$. In particular,
two vertices are joined by an edge whenever their labels differ on a
permutation of two neighbor elements. The complex ${CP}$ cannot be represented
by a convex polytope, since it is not a combinatorial sphere (not even a
combinatorial manifold). However, it can be represented by some \textit{virtual
polytope} (Minkowski difference of two convex polytopes) which we call
"cyclopermutohedron" $\mathcal{CP}_{n+1}$. It is defined explicitly, as a
weighted Minkowski sum of line segments. Informally, the cyclopermutohedron can
be viewed as "permutohedron with diagonals". One of the motivations is that the
cyclopermutohedron is a "universal" polytope for moduli spaces of polygonal
linkages.
