Using data from contemporaneous observations with Chandra and Swift, it is shown that the X-ray emission below 10 keV is predominantly thermal, characterized by flares and emission lines and dominated by the soft component. The Chandra and Swift X-ray spectra (EX≤ 10 keV) can be reproduced by multicomponent thermal emission models with a time-averaged X-ray luminosity of LX∼ 1031 erg s−1. The pulsed 33-s soft X-ray emission below 10 keV is confirmed in both the Chandra and Swift data sets. The epoch of pulse maximum of the 33-s white dwarf spin period is consistent with the recently derived ephemeris based upon Suzaku measurements. The recently detected Suzaku hard X-ray component above 10 keV shows a non-thermal power-law nature, with a photon index of Γ∼ 1.2, possibly the result of synchrotron emission of high-energy electrons in the white dwarf magnetosphere. The hard X-ray luminosity of LX,hard≤ 5 × 1030 erg s−1 also constitutes κ∼ 0.1 per cent of the total spin-down luminosity of the white dwarf. This places AE Aquarii in the same category as young spin-powered pulsars between 2 and 20 keV. Additionally, it is shown that electrons can be accelerated to energies in excess of 10 TeV outside the light cylinder radius, providing interesting possibilities for VHE-TeV follow-up observations. The X-ray emission below EX≤ 10 keV, on the other hand, is explained in terms of plasma heating at the magnetospheric radius, the result of the dissipation of gravitational potential energy. It is found that a conversion efficiency of α∼ 0.01 is sufficient to heat the plasma at the magnetospheric boundary to temperatures kT≤ 10 keV, sufficient to drive the X-ray emission below 10 keV. Only a small fraction (β∼ 0.3 per cent) of the mass flow at the magnetospheric radius eventually accretes on to the surface of the white dwarf, emphasizing the very effective magnetospheric propeller process in the system.