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Usually, photons are described by plane waves with a definite 4-momentum. In addition to plane-wave photons, "twisted photons" have recently entered the field of modern laser optics; these are coherent superpositions of plane waves with a defined projection hm of the orbital angular momentum onto the propagation axis, where m is an integer. In this Letter, we show that it is possible to produce high-energy twisted photons by Compton backscattering of twisted laser photons off ultrarelativistic electrons. Such photons may be of interest for experiments related to the excitation and disintegration of atoms and nuclei, and for studying the photoeffect and pair production off nuclei in previously unexplored experimental regimes.