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

Schur-Weyl duality and the Product of randomly-rotated symmetries by a unitary Brownian motion