  There are two frameworks for mating Kleinian groups with rational maps on the
Riemann sphere: the algebraic correspondence framework due to
Bullett-Penrose-Lomonaco \cite{BP94,BL20a} and the simultaneous uniformization
mating framework of \cite{MM1}. The current paper unifies and generalizes these
two frameworks in the case of principal hyperbolic components. To achieve this,
we extend the mating framework of \cite{MM1} to genus zero hyperbolic orbifolds
with at most one orbifold point of order $\nu \geq 3$ and at most one orbifold
point of order two. We give an explicit description of the resulting conformal
matings in terms of uniformizing rational maps. Using these rational maps, we
construct correspondences that are matings of such hyperbolic orbifold groups
(including the modular group) with polynomials in principal hyperbolic
components. We also define an algebraic parameter space of correspondences and
construct an analog of a Bers slice of the above orbifolds in this parameter
space.
