We have used ISO to observe the Magnetic Cataclysmic Variable AE Aquarii in the previously unexplored range from 4.8 $\mu$m up to 170 $\mu$m in the framework of a coordinated multi-wavelength campaign from the radio to optical wavelengths. We have obtained for the first time a spectrum between 4.8 and 7.3 $\mu$m with ISOCAM and ISOPHOT-P: the major contribution comes from the secondary star spectrum, with some thermal emission from the accretion stream, and possibly some additional cyclotron radiation from the post-shock accretion material close to the magnetised white dwarf. Having reprocessed ISOPHOT-C data, we confirm AE Aqr detection at $90~\mu$m and we have re-estimated its upper limit at 170 $\mu$m. In addition, having re-processed IRAS data, we have detected AE Aqr at 60 $\mu$m and we have estimated its upper limits at 12, 25, and 100 $\mu$m. The literature shows that the time-averaged spectrum of AE Aqr increases roughly with frequency from the radio wavelengths up to ${\sim} 761~ \mu$m; our results indicate that it seems to be approximately flat between ~761 and ${\sim} 90 ~\mu$m, at the same level as the 3$\sigma$ upper limit at 170 $\mu$m; and it then decreases from ${\sim} 90 ~\mu$m to ${\sim} 7~ \mu$m. Thermal emission from dust grains or from a circum-binary disc seems to be very unlikely in AE Aqr, unless such a disc has properties substantially different from those predicted recently. Since various measurements and the usual assumptions on the source size suggest a brightness temperature below 109 K at $\lambda \leq 3.4$ mm, we have reconsidered also the possible mechanisms explaining the emission already known from the submillimetre to the radio. The complex average spectrum measured from ${\sim} 7~ \mu$m to the radio must be explained by emission from a plasma composed of more than one "pure" non-thermal electron energy distribution (usually assumed to be a power-law): either a very large volume (diameter $\geq$ 80 times the binary separation) could be the source of thermal bremsstrahlung which would dominate from ${\sim} 10 ~\mu$m to the ~millimetre, with, inside, a non-thermal source of synchrotron which dominates in radio; or, more probably, an initially small infrared source composed of several distributions (possibly both thermal, and non-thermal, mildly relativistic electrons) radiates gyro-synchrotron and expands moderately: it requires to be re-energised in order to lead to the observed, larger, radio source of highly relativistic electrons (in the form of several non-thermal distributions) which produce synchrotron.