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Context. Edge-on galaxies can offer important insights in galaxy evolution as they are the only systems where the distribution of the different components can be studied both radially and vertically. The HEROES project was designed to investigate the interplay between the gas, dust, stars and dark matter (DM) in a sample of 7 massive edge-on spiral galaxies. Aims. In this second HEROES paper we present an analysis of the atomic gas content of 6 out of 7 galaxies in our sample. The remaining galaxy was recently analysed according to the same strategy. The primary aim of this work is to constrain the surface density distribution, the rotation curve and the geometry of the gas disks in a homogeneous way. In addition we identify peculiar features and signs of recent interactions. Methods. We construct detailed tilted-ring models of the atomic gas disks based on new GMRT 21-cm observations of NGC 973 and UGC 4277 and re-reduced archival HI data of NGC 5907, NGC 5529, IC 2531 and NGC 4217. Potential degeneracies between different models are resolved by requiring a good agreement with the data in various representations of the data cubes. Results. From our modelling we find that all but one galaxy are warped along the major axis. In addition, we identify warps along the line of sight in three galaxies. A flaring gas layer is required to reproduce the data only for one galaxy, but (moderate) flares cannot be ruled for the other galaxies either. A coplanar ring-like structure is detected outside the main disk of NGC 4217, which we suggest could be the remnant of a recent minor merger event. We also find evidence for a radial inflow of 15 +- 5 km/s in the disk of NGC 5529, which might be related to the ongoing interaction with two nearby companions. (Abridged), 39 pages, 38 figures, Accepted for publication in Astronomy & Astrophysics