We treat Killing-transversally symmetric spaces (briclly. KTS-spaces), that is Riemannian manifolds equipped with a complete unit Killing vector field such that the reflections with respect to the flow lines of that field can be extended to global isometries. Sucha manifolds are homogencous spaces equipped with a naturally reductive homogeneous structure and they provide a rich set of examples of reflection spaces. We prove that cach simply connected reducible KTS-space M is a Ricmannian product of a symmetric space M\u27 and a special kind of KTS-space M\u27\u27, called a contact KTS-space. Such a particular manifold M\u27\u27 is an irreducible, odd-dimensional principal G-bundle over a Hermitian symmectric space. The main purpose of the paper is to give a classification of this special class of manifolds M\u27\u27
