Through their delivery of water and organics, near-Earth objects (NEOs) played an important role in the emergence of life on our planet. However, they also pose a hazard to the Earth, as asteroid impacts could significantly affect our civilization. Potentially hazardous asteroids (PHAs) are those that, in principle, could possibly impact the Earth within the next century, producing major damage. About 1600 PHAs are currently known, from an estimated population of 4700±1450. However, a comprehensive characterization of the PHA physical properties is still missing. Here we present spectroscopic observations of 14 PHAs, which we have used to derive their taxonomy, meteorite analogs, and mineralogy. Combining our results with the literature, we investigated how PHAs are distributed as a function of their dynamical and physical properties. In general, the "carbonaceous" PHAs seem to be particularly threatening, because of their high porosity (limiting the effectiveness of the main deflection techniques that could be used in space) and low inclination and minimum orbit intersection distance (MOID) with the Earth (favoring more frequent close approaches). V-type PHAs also present low MOID values, which can produce frequent close approaches (as confirmed by the recent discovery of a limited space weathering on their surfaces). We also identified those specific objects that deserve particular attention because of their extreme rotational properties, internal strength, or possible cometary nature. For PHAs and NEOs in general, we identified a possible anti-correlation between the elongation and the rotational period, in the range of P rot ≈5-80 hr. This would be compatible with the behavior of gravity-dominated aggregates in rotational equilibrium. For periods 80-90 hr, such a trend stops, possibly under the influence of the YORP effect and collisions. However, the statistics is very low, and further observational and theoretical work is required to characterize such slow rotators.