[Abridged] We investigated the X-ray characteristics of the Class I YSO Elias 29 with joint XMM-Newton and NuSTAR observations of 300 ks and 450 ks, respectively. These are the first observations of a very young (<1 Myr) stellar object in a band encompassing simultaneously both soft and hard X-rays. In addition to the hot Fe complex at 6.7 keV, we observed fluorescent emission from Fe at $sim6.4$ keV, confirming the previous findings. The line at 6.4 keV is detected during quiescent and flaring states and its flux is variable. The equivalent width is found varying in the $approx 0.15--0.5$ keV range. These values make unrealistic a simple model with a centrally illuminated disk and suggest a role of the cavity containing Elias 29 and possibly reverberation processes that could occur in it. We observed two flares, with duration of 20 ks and 50 ks, respectively. We systematically observed an increase of $N_H$ during the flares of a factor five. This behavior has been observed during flares previously detected in Elias 29 with XMM-Newton and ASCA. The phenomenon hints that the flaring regions could be buried under the accretion streams and at high stellar latitudes, as the X-rays from flares pass through gas denser than the gas along the line of sight of the quiescent corona. In a different scenario, a contribution from scattered soft photons to the primary coronal emission could mimic a shallower $N_H$ in the quiescent spectrum. In the spectrum of the full NuSTAR exposure, we detect hard X-ray emission in the band $approx20-80$ keV in excess with respect to the thermal emission. The hard X-ray emission could be due to a population of energetic electrons accelerated by the magnetic field along the accretion streams. These particles could concur to pumping up the Fe fluorescence of cold Fe of the disk along with X-ray photons with $E>7.11$ keV.
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