  Orthogonal rational functions (ORF) on the unit circle generalize orthogonal
polynomials (poles at infinity) and Laurent polynomials (poles at zero and
infinity). In this paper we investigate the properties of and the relation
between these ORF when the poles are all outside or all inside the unit disk,
or when they can be anywhere in the extended complex plane outside the unit
circle. Some properties of matrices that are the product of elementary unitary
transformations will be proved and some connections with related algorithms for
direct and inverse eigenvalue problems will be explained.
