  A well known quotient of the real Stiefel manifold is the projective Stiefel
manifold. We introduce a new family of quotients of the real Stiefel manifold
by cyclic group of order 2 whose action is induced by simultaneous pairwise
flipping of the coordinates. We obtain a description for their tangent bundles,
compute their mod 2 cohomology and compute Stiefel Whitney classes of these
manifolds. We use these to give applications to their stable span,
parallelizability and equivariant maps, and the associated results in
topological combinatorics.
