We report on XMM-Newton and NuSTAR X-ray observations of the prototypical polar, AM Herculis, supported by ground-based photometry and spectroscopy, all obtained in high accretion states. In 2005, AM Herculis was in its regular mode of accretion, showing a self-eclipse of the main accreting pole. X-ray emission during the self-eclipse was assigned to a second pole through its soft X-ray emission and not to scattering. In 2015, AM Herculis was in its reversed mode with strong soft blobby accretion at the far accretion region. The blobby acretion region was more luminous than the other, persistently accreting, therefore called main region. Hard X-rays from the main region did not show a self-eclipse indicating a pronounced migration of the accretion footpoint. Extended phases of soft X-ray extinction through absorption in interbinary matter were observed for the first time in AM Herculis. The spectral parameters of a large number of individual soft flares could be derived. Simultaneous NuSTAR observations in the reversed mode of accretion revealed clear evidence for Compton reflection of radiation from the main pole at the white dwarf surface. This picture is supported by the trace of the Fe resonance line at 6.4 keV through the whole orbit. Highly ionized oxygen lines observed with the Reflection Grating Spectrometer (RGS) were tentatively located at the bottom of the accretion column, although the implied densities are quite different from expectations. In the regular mode of accretion, the phase-dependent modulations in the ultraviolet (UV) are explained with projection effects of an accretion-heated spot at the prime pole. In the reversed mode projection effects cannot be recognized. The light curves reveal an extra source of UV radiation and extended UV absorbing dips. An Ha Doppler map obtained contemporaneously (abstract abridged)
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