  Cone spherical surfaces are orientable Riemannian surfaces with constant
curvature one and a finite set of conical singularities. A subset of these
surfaces, referred to as dihedral surfaces, is characterized by their monodromy
groups, which notably preserve a pair of antipodal points on the unit
two-sphere within three-dimensional Euclidean space. On each dihedral surface,
we define a pair of transverse measured foliations that, in turn,
comprehensively characterize the original dihedral surface. Furthermore, we
introduce a variety of geometric decompositions and deformations specific to
dihedral surfaces. As a practical application, we ascertain the dimension of
the moduli space for dihedral surfaces given specified cone angles and
topological types. This dimension acts as an indicator of the independent
geometric parameters that determine the isometric classes of these surfaces.
