  We glue two families of Bernstein-Szego polynomials to construct the
eigenbasis of an associated finite-dimensional Jacobi matrix. This gives rise
to finite orthogonality relations for this composite eigenbasis of
Bernstein-Szego polynomials. As an application, a number of Gauss-like
quadrature rules are derived for the exact integration of rational functions
with prescribed poles against the Chebyshev weight functions.
