  The efficacy of numerical methods like integral estimates via Gaussian
quadrature formulas depends on the localization of the zeros of the associated
family of orthogonal polynomials. In this regard, following the renewed
interest in quadrature formulas on the unit circle, and $R_{II}$-type
polynomials, which include the complementary Romanovski-Routh polynomials, in
this work we present a collection of properties of their zeros. Our results
include extreme bounds, convexity, and density, alongside the connection of
such polynomials to classical orthogonal polynomials via asymptotic formulas.
