We investigated the time-averaged high-energy spectral characteristics of the persistent anomalous X-ray pulsars (AXPs) 1RXS J1708-4009, 4U 0142+61, 1E 2259+586, and 1E 1048.1-5937, using RXTE PCA (2-60 keV), RXTE HEXTE (15-250 keV), and INTEGRAL IBIS ISGRI (20-300 keV) data. We discovered hard spectral tails for energies above 10 keV in the total and pulsed spectra of AXPs 1RXS J1708-4009, 4U 0142+4009, and 1E 2259+586, but 1E 1048.1-5937 appeared to be too weak to be detected. Improved hard X-ray spectral information for AXP 1E 1841-045 is also presented. The pulsed and total spectra above 10 keV have power-law shapes, and there is so far no significant evidence for spectral breaks or bends up to ~150 keV. The pulsed spectra above 10 keV are exceptionally hard with indices for four AXPs in the range -1.0 to 1.0. Below 10 keV these indices were in the range 2.0-4.3, indicating the very drastic spectral changes in a narrow energy interval around 10 keV. The best-fit power-law models to the total spectra between ~10 and 150 keV are significantly softer, with indices measured for 1E 1841-045, 1RXS J1708-4009, and 4U 0142+61 in the range 1.0-1.4. For the latter AXPs the pulsed fractions are consistent with 100% around 100 keV but are different at 10 keV: ~10% for 4U 0142+61, ~25% for 1E 1841-045, and consistent with 100% for 1RXS J1708-4009. The luminosities of these total and pulsed spectral tails (10-150 keV) largely exceed the total available spin-down powers by factors ranging from ~100 to ~600. We also reanalyzed archival CGRO COMPTEL (0.75-30 MeV) data: no AXP detections can be claimed, and the obtained upper limits indicate for 1RXS J1708-4009, 4U 0142+61, and 1E 1841-045 that strong breaks or bends must occur somewhere between ~150 and 750 keV. We discuss predictions from first attempts to model our hard X-ray and soft gamma-ray spectra in the context of the magnetar model.
