  We study the perspectives of measuring the phenomenon of vacuum birefringence
predicted by quantum electrodynamics using an x-ray free-electron laser (XFEL)
alone. We devise an experimental scheme allowing the XFEL beam to collide with
itself under a finite angle, and thus act as both pump and probe field for the
effect. The signature of vacuum birefringence is encoded in
polarization-flipped signal photons to be detected with high-purity x-ray
polarimetry. Our findings for idealized scenarios underline that the discovery
potential of solely XFEL-based setups can be comparable to those involving
optical high-intensity lasers. For currently achievable scenarios, we identify
several key details of the x-ray optical ingredients that exert a strong
influence on the magnitude of the desired signatures.
