In this paper, we present the hard-X-ray spectral analysis of a complete sample of active galactic nuclei (AGNs) detected by INTEGRAL/IBIS. In conjunction with IBIS spectra, we make use of Swift/BAT data, with the aim of cross-calibrating the two instruments, studying source variability and constraining some important spectral parameters. We find that flux variability is present in at least 14 per cent of the sample, while spectral variability is found only in one object. There is general good agreement between BAT and IBIS spectra, despite a systematic mismatch of about 22 per cent in normalization. When fitted with a simple power-law model, type 1 and type 2 sources appear to have very similar average photon indices, suggesting that they are powered by the same mechanism. As expected, we also find that a simple power law does not always describe the data sufficiently well, thus indicating a certain degree of spectral complexity, which can be ascribed to features like a high energy cut-off and/or a reflection component. Fixing the reflection to be 0, 1 or 2, we find that our sample covers quite a large range in photon indices as well as cut-off energies; however, the spread is due only to a small number of objects, while the majority of the AGNs lie within well-defined boundaries of photon index (1 ≤ Γ ≤ 2) and cut-off energy (30 ≤ E cut ≤ 300 keV).