  In this paper we illustrate that paraorthogonality on the unit circle
$\mathbb{T}$ is the counterpart to orthogonality on $\mathbb{R}$ when we are
interested in the spectral properties. We characterize quasi-paraorthogonal
polynomials on the unit circle as the analogues of the quasi-orthogonal
polynomials on $\mathbb{R}$. We analyze the possibilities of preselecting some
of its zeros, in order to build positive quadrature formulas with prefixed
nodes and maximal domain of validity. These quadrature formulas on the unit
circle are illustrated numerically.
