  The study of the set-theoretic solutions of the reflection equation, also
known as reflection maps, is closely related to that of the Yang-Baxter maps.
In this work, we construct reflection maps on various geometrical objects,
associated with factorization problems on rational loop groups and involutions.
We show that such reflection maps are smoothly conjugate to the composite of
permutation maps, with corresponding reduced Yang-Baxter maps. In the case when
the reduced Yang-Baxter maps are independent of parameters, the latter are just
braiding operators. We also study the symplectic and Poisson geometry of such
reflection maps. In a special case, the factorization problems are associated
with the collision of N-solitons of the n-Manakov system with a boundary, and
in this context the N-body polarization reflection map is a symplectomorphism.
