  We study averages of multiplicative eigenvalue statistics in ensembles of
orthogonal Haar distributed matrices, which can alternatively be written as
Toeplitz+Hankel determinants. We obtain new asymptotics for symbols with
Fisher-Hartwig singularities in cases where some of the singularities merge
together, and for symbols with a gap or an emerging gap. We obtain these
asymptotics by relying on known analogous results in the unitary group and on
asymptotics for associated orthogonal polynomials on the unit circle. As
consequences of our results, we derive asymptotics for gap probabilities in the
Circular Orthogonal and Symplectic Ensembles, and an upper bound for the global
eigenvalue rigidity in the orthogonal ensembles.
