The Los Alamos hybrid simulation code is used to examine kinetic properties of pickup ions at the heliospheric termination shock and in the downstream heliosheath. All simulations are one‐dimensional in spatial variations, represent the electrons as a zero‐mass fluid, and address only perpendicular shocks. Interpretation of measurements from the IBEX and Voyager spacecraft depend sensitively on the properties of the suprathermal ions downstream of the termination shock, so this research addresses three topics concerning such ions. First, a careful examination of pickup ion trajectories shows that their initial acceleration does not require specular reflection at the shock, as is sometimes assumed, but is the consequence of gyromotion by selected ions at the shock. The primary factor in this energy gain is a gyro‐phase‐dependent interaction with the motional electric field upstream of, and the magnetic field at, the shock. Second, shock simulations are carried out in which the upstream pickup ions are assumed to have four different types of velocity distributions. The downstream ion perpendicular velocity distributions f(v⊥) are similar in each of the runs and may be approximately characterized as a thermal Maxwellian and a suprathermal distribution. The only significant difference among the four downstream distributions is in the tails of the suprathermal component. Third, simulations are carried out for three different upstream Mach numbers; the results show that faster solar wind flows lead to increased fluxes of ions in the tails of the suprathermal component and are generally consistent with energetic neutral atom observations by the IBEX spacecraft.