We describe and study an explicit structure of a regular cell complex K(L)K(L) on the moduli space M(L) of a planar polygonal linkage L. The combinatorics is very much related (but not equal) to the combinatorics of the permutohedron. In particular, the cells of maximal dimension are labeled by elements of the symmetric group. For example, if the moduli space M is a sphere, the complex KK is dual to the boundary complex of the permutohedron.The dual complex K∗K∗ is patched of Cartesian products of permutohedra. It can be explicitly realized in the Euclidean space via a surgery on the permutohedron.