Authors’ abstract: Motivated by the works of P. Delsarte and Y. Genin [SIAM J. Math. Anal. 19, No. 3, 718–735 (1988; Zbl 0638.30037); SIAM J. Matrix Anal. Appl. 12, No. 2, 220–238 (1991; Zbl 0728.65020); ibid. 12, No. 3, 432–448 (1991; Zbl 0753.15015)], who studied paraorthogonal polynomials associated with positive definite Hermitian linear functionals and their corresponding recurrence relations, we provide paraorthogonality theory, in the context of quasidefinite Hermitian functionals, with a recurrence relation and the analogous result to the classical Favard’s theorem or spectral theorem. As an application of our results, we prove that for any two monic polynomials whose zeros are simple and strictly interlacing on the unit circle, with the possible exception of one of them which could be common, there exists a sequence of paraorthogonal polynomials such that these polynomials belong to it. Furthermore, an application to the computation of the Szegő quadrature formulas is also discussed.