We present a deep study of the average hard X-ray spectra of Seyfert galaxies . We aim to test the unified model of active galactic nuclei , and constrain differences and similarities between different classes of objects . We analyzed all public INTEGRAL IBIS/ISGRI data available on all the 165 Seyfert galaxies detected at z < 0.2 . Our final sample consists of 44 Seyfert 1s , 29 Seyfert 1.5s , 78 Seyfert 2s , and 14 narrow-line Seyfert 1s . For each subsample , we stacked all the images , and derived their average hard X-ray spectra in the 17–250 keV energy range . We performed a detailed spectral analysis using both a model-independent and a model-dependent approach . All classes of Seyfert galaxies show on average the same nuclear continuum , as foreseen by the zeroth order unified model , with a cutoff energy of E _ { C } \gtrsim 200 keV , and a photon index of \Gamma \simeq 1.8 . The average optical depth of the Comptonizing medium is consistent for the different classes ( \tau \simeq 0.8 ) . Compton-thin Seyfert 2s show a reflection component stronger than Seyfert 1s and Seyfert 1.5s . Most of this reflection is due to mildly obscured ( 10 ^ { 23 } cm ^ { -2 } \leq N _ { H } < 10 ^ { 24 } cm ^ { -2 } ) Seyfert 2s , which have a significantly stronger reflection component ( R = 2.2 ^ { +4.5 } _ { -1.1 } ) than Seyfert 1s ( R \leq 0.4 ) , Seyfert 1.5s ( R \leq 0.4 ) and lightly obscured ( N _ { H } < 10 ^ { 23 } cm ^ { -2 } ) Seyfert 2s ( R \leq 0.5 ) . This can not be explained easily by the unified model . The absorber/reflector in mildly obscured Seyfert 2s might cover a large fraction of the X-ray source , and contain clumps of Compton-thick material . The large reflection found in the spectrum of mildly obscured Seyfert 2s reduces the amount of Compton-thick objects needed to explain the peak of the cosmic X-ray background . Our results are consistent with the fraction of Compton-thick sources being \sim 10 \% . The spectra of Seyfert 2s with and without polarized broad lines do not show significant differences , the only difference between the two samples being the higher hard X-ray and bolometric luminosity of Seyfert 2s with polarized broad lines . The average hard X-ray spectrum of narrow-line Seyfert 1s is steeper than those of Seyfert 1s and Seyfert 1.5s , probably due to a lower energy of the cutoff .