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The spectroscopic response of CdTe detectors has been investigated with an unusual irradiation geometry, i.e., with the beam impinging at various angles through the cathode surface. In fact, the classical planar parallel field (PPF) irradiation geometry, i.e., with photons impinging perpendicularly to the cathode, limits the spectroscopic performance at high energies (E>60 keV), if thick detectors are used, owing to the large spread in the paths of the charge carriers. These paths for the holes can be greatly reduced by using larger impinging angles, thus improving the energy resolution without losses in detection efficiency. Several sets of measurements have been carried out with a narrow photon beam, coming from collimated 57Co, 109Cd and 241Am sources; measurement angles ranged from 0° (orthogonal to the cathode) to 87° (beam grazing the cathode). The spectroscopic responses of the detectors for 14, 22, 60 and 122 keV photon energies have been analyzed. Detectors equipped with both full anodes and microstrip anodes were used. In this last case the rectangular collimated spot of the beam was used with the long side both parallel and orthogonal to the strip direction. The experimental results show that an actual improvement in the spectroscopic performance is obtained, mostly for the high-energy X-rays, by this tilting procedure. Further, this technique allows the evaluation of the detector dead layer thickness. The obtained performance and possible medical and space applications are discussed.