  Motivated by the work of Panina and her coauthors on cyclopermutohedron we
study a poset whose elements correspond to equivalence classes of partitions of
the set $\{1,\cdots, n+1\}$ up to cyclic permutations and orientation
reversion. This poset is the face poset of a regular CW complex which we call
bi-cyclopermutohedron and denote it by $\mathrm{QP}_{n+1}$. The complex
$\mathrm{QP}_{n+1}$ contains subcomplexes homeomorphic to moduli space of
certain planar polygons with $n+1$ sides up to isometries. In this article we
find an optimal discrete Morse function on $\mathrm{QP}_{n+1}$ and use it to
compute its homology with $\mathbb{Z}$ as well as $\mathbb{Z}_2$ coefficients.
