  In this paper, we introduce and study a unitary matrix-valued process which
is closely related to the Hermitian matrix-Jacobi process. It is precisely
defined as the product of a deterministic self-adjoint symmetry and a
randomly-rotated one by a unitary Brownian motion. Using stochastic calculus
and the action of the symmetric group on tensor powers, we derive an autonomous
ordinary differential equation for the moments of its fixed-time marginals.
Next, we derive an expression of these moments which involves a unitary bridge
between our unitary process and another independent unitary Brownian motion.
This bridge motivates and allows to write a second direct proof of the obtained
moment expression.
