By analysing a sample of galaxies selected from the HI Parkes All Sky Survey (HIPASS) to contain more than 2.5 times their expected HI content based on their optical properties, we investigate what drives these HI eXtreme (HIX) galaxies to be so HI-rich. We model the HI kinematics with the Tilted Ring Fitting Code TiRiFiC and compare the observed HIX galaxies to a control sample of galaxies from HIPASS as well as simulated galaxies built with the semianalytic model DARK SAGE. We find that (1) HI discs in HIX galaxies are more likely to be warped and more likely to host HI arms and tails than in the control galaxies, (2) the average HI and average stellar column density of HIX galaxies is comparable to the control sample, (3) HIX galaxies have higher HI and baryonic specific angular momenta than control galaxies, (4) most HIX galaxies live in higher spin haloes than most control galaxies. These results suggest that HIX galaxies are HI-rich because they can support more HI against gravitational instability due to their high specific angular momentum. The majority of the HIX galaxies inherits their high specific angular momentum from their halo. The HI content of HIX galaxies might be further increased by gas-rich minor mergers. This paper is based on data obtained with the Australia Telescope Compact Array through the large program C 2705.