  We prove and explore a family of identities relating lengths of curves and
orthogeodesics of hyperbolic surfaces. These identities hold over a large space
of metrics including ones with hyperbolic cone points, and in particular, show
how to extend a result of the first author to surfaces with cusps. One of the
main ingredients in the approach is a partition of the set of orthogeodesics
into sets depending on their dynamical behavior, which can be understood
geometrically by relating them to geodesics on orbifold surfaces. These
orbifold surfaces turn out to be exactly on the boundary of the space in which
the underlying identity holds.
