I think your best course of action would be actually to write a fully fledged compositor.
There is the GLX_texture_from_pixmap
extension, that allows you to source any pixmap compatible X11 drawable into a OpenGL texture. For a start it might be enough to simply pull the root window (pixmap) as it is into a OpenGL texture. Later you might want to use the Composite extension to redirect windows to off-screen rendering and composite them in 3D space as a stereoscopic picture in the Occulus Rift.