In this paper we report on the broad-band X-ray properties of a complete sample of absorbed Seyfert galaxies hard X-ray selected with INTEGRAL. Our sample is composed of 33 sources, of which 15 are newly discovered active galactic nuclei (AGN) above 20 keV (IGR sources), while 18 are already known type 2 AGN (‘known’). For 17 sources (15 IGR + 2 ‘known’ sources) we have performed a broad-band analysis using both XMM–Newton, and INTEGRAL-IBIS data. To have a full view of the complete sample we have then complemented the analysis of the 16 remaining sources with already existing broad-band studies in the same range. The high-quality broad-band spectra are well reproduced with an absorbed primary emission with a high-energy cut-off and its scattered fraction below 2–3 keV, plus the Compton reflection features (Compton hump and Fe line emission). This study permitted a very good characterization of the primary continuum and, in turn, of all the spectral features.
A high-energy cut-off is found in 30 per cent of the sample, with an average value below 150 keV, suggesting that this feature has to be present in the X-ray spectra of obscured AGN. The hard X-ray selection favours the detection of more obscured sources, with the log NH average value of 23.15 (standard deviation of 0.89). The diagnostic plot NH versus Foss(2–10 keV)/F(20–100 keV) allowed the isolation of the Compton-thick objects, and may represent a useful tool for future hard X-ray observations of newly discovered AGN. We are unable to associate the reflection components (both continuum and Fe line) with the absorbing gas as a torus (as envisaged in the Unified Model), a more complex scenario being necessary. In the Compton-thin sources, a fraction (but not all) of the Fe K line needs to be produced in a gas located closer to the black hole than the Compton-thick torus, and this is possibly associated with the optical broad-line region, responsible also for the absorption. We still need a Compton-thick medium (not intercepting the line of sight) likely associated to a torus, which contributes to the Fe line intensity and produces the observed reflection continuum above 10 keV. The so-called Iwasawa–Taniguchi effect cannot be confirmed with our data. Finally, the comparison with a sample of unobscured AGN shows that type 1 and type 2 (once corrected for absorption) Seyfert are characterized by the same nuclear/accretion properties (luminosity, bolometric luminosity, Eddington ratio), supporting the ‘unified’ view.