  In this paper we study quasi-orthogonality on the unit circle based on the
structural and orthogonal properties of a class of self-invariant polynomials.
We discuss a special case in which these polynomials are represented in terms
of the reversed Szeg\H{o} polynomials of consecutive degrees and illustrate the
results using contiguous relations of hypergeometric functions. This work is
motivated partly by the fact that recently cases have been made to establish
para-orthogonal polynomials as the unit circle analogues of quasi-orthogonal
polynomials on the real line so far as spectral properties are concerned. We
show that structure wise too there is great analogy when self-inversive
polynomials are used to study quasi-orthogonality on the unit circle.
