The $n$-associahedron is a well-known $n$-dimensional polytope whose vertices are labeled by triangulations of an $(n+3)$-gon with edges given by diagonal flips. The $n$-cyclohedron is defined analogously using centrally-symmetric triangulations of a $(2n+2)$-gon, or, modding out by the symmetry, triangulations of an $(n+1)$-gon with one orbifold point.  The polytopes can be realized in such a way that their normal fans are the “$\mathbf{g}$-vector fans”, or “mutation fans”, for certain cluster algebras. In this talk I will justify and generalize two relationships which hold between these fans: the normal fan to the $n$-cyclohedron can be obtained by intersecting the normal fan to the $(2n-1)$-associahedron with a certain subspace, and the normal fan to the $n$-cyclohedron refines the normal fan to the $n$-associahedron. I will show that these relationships are instances of “folding” and “dominance,” respectively, and hold more generally for mutation fans for cluster algebras modeled by surfaces and orbifolds.