  We define symmetric bundles as vector bundles in the category of symmetric
spaces; it is shown that this notion is the geometric analog of the one of a
representation of a Lie triple system. We show that such a bundle has an
underlying reflection space, and we investigate the corresponding forgetful
functor both from the point of view of differential geometry and from the point
of view of representation theory. This functor is not injective, as is seen by
constructing "unusual" symmetric bundle structures on the tangent bundles of
certain symmetric spaces.
