We present theoretical considerations for diffracted diffraction radiation and also propose an application of this process to diagnosing ultra-relativistic electron (positron) beams for the first time. Diffraction radiation is produced when relativistic particles move near a target. If the target is a crystal or X-ray mirror, diffraction radiation in the X-ray region is expected to be diffracted at the Bragg angle and therefore be detectable. We present a scheme for applying this process to measurements of the beam angular spread, and consider how to conduct a proof-of-principle experiment for the proposed method.