  We generalize a new class of cluster type mutations for which exchange
transformations are given by reciprocal polynomials. In the case of
second-order polynomials of the form $x+2\cos{\pi/n_o}+x^{-1}$ these
transformations are related to triangulations of Riemann surfaces of arbitrary
genus with at least one hole/puncture and with an arbitrary number of orbifold
points of arbitrary integer orders $n_o$. We propose the dual graph description
of the corresponding Teichm\"uller spaces, construct the Poisson algebra of the
Teichm\"uller space coordinates, propose the combinatorial description of the
corresponding geodesic functions and find the mapping class group
transformations.
