The atomic-level mechanism of the reaction of H atoms with triplet and singlet molecular oxygen, H(S-2) + O-2((3)Sigma(-)(g)) -> O-2(P-3) + OH((2)Pi(g)) (R1) and H(S-2) + O-2((1)Delta(g)) -> O(P-3) + OH((2)Pi(g)) (R2) is analyzed in terms of the topology of the potential energy surfaces (PES) of the two reactions. Both PES exhibit a deep potential well corresponding to the ground and first excited electronic state of HO2. The ground-state reaction is endothermic with no barrier on either side of the well; the excited-state reaction is exothermic with a barrier in the entrance valley of the PES. The differences of the PES are manifested in properties such as the excitation functions, which show reaction R1 to be much slower and the effect of rotational excitation on reactivity, which speeds up reaction R1 and has little effect on R2. Numerous common dynamics features arise from the presence of the deep potential well on the PES. Such are the significant role of isomerization (for example, 90% of reactive collisions in R2 involve at least one H atom transfer from one of the O atoms to the other in reaction R2), which is shown to give rise to a significant rotational excitation of the product significant sideways scattering of the products that originates from collisions in propeller-type arrangements induced by the presence of two bands of acceptance around the O-2 molecule. The HO2 complex in both reactions proves to behave nonstatistically, with signatures of the dynamics in lifetime distributions, angular distributions, opacity functions, and product quantum-state distributions.