We determine the intrinsic shapes and orientations of 27 450 types I and II active galactic nucleus (AGN) galaxies in the spectroscopic sample of the Sloan Digital Sky Survey Data Release 7 by studying the distribution of projected axial ratios of AGN hosts. Our aim is to study possible alignments between the AGN and host galaxy systems (e.g. the accretion disc and the galaxy angular momentum) and the effect of dust obscuration geometry on the AGN type. We define control samples of non-AGN galaxies that mimic the morphology, colour, luminosity and concentration distributions of the AGN population, taking into account the effects of dust extinction and reddening. Assuming that AGN galaxies have the same underlying 3D shape distribution as their corresponding control samples, we find that the spiral and elliptical type I AGN populations are strongly skewed towards face-on galaxies, while elliptical and spiral type II AGN populations are skewed towards edge-on orientations. These findings rule out random orientations for AGN hosts at high confidence for type I spirals (δχ2≈ 230) and type II ellipticals (δχ2≈ 15), while the signal for type I ellipticals and type II spirals is weaker (δχ2≈ 3 and ≈6, respectively). We obtain a much stronger tendency for the type II spirals to be edge-on when just high [O iii] equivalent width (EW) AGN are considered, suggesting that >20 per cent of low [O iii] EW edge-on type II AGN may be missing from the optical sample. Galactic dust absorption of the broad-line region alone cannot explain the observed inclination angle and projected axial ratio distributions of types I and II Seyfert galaxies, implying that obscuration by a small-scale circumnuclear torus is necessary. These results favour a scenario in which the angular momentum of the material which feeds the black hole retains a memory of its original gas source at least to some small, non-negligible degree.