  In this paper, we continue exploration of the dynamical and parameter planes
of one-parameter families of Schwarz reflections that was initiated in
\cite{LLMM1,LLMM2}. Namely, we consider a family of quadrature domains obtained
by restricting the Chebyshev cubic polynomial to various univalent discs. Then
we perform a quasiconformal surgery that turns these reflections to parabolic
rational maps (which is the crucial technical ingredient of our theory). It
induces a straightening map between the parameter plane of Schwarz reflections
and the parabolic Tricorn. We describe various properties of this straightening
highlighting the issues related to its anti-holomorphic nature. We complete the
discussion by comparing our family with the classical Bullett-Penrose family of
matings between groups and rational maps induced by holomorphic
correspondences. More precisely, we show that the Schwarz reflections give rise
to anti-holomorphic correspondences that are matings of parabolic anti-rational
maps with the abstract modular group. We further illustrate our mating
framework by studying the correspondence associated with the Schwarz reflection
map of a deltoid.
