We calculate cross sections for helium-antihydrogen scattering for energies up to 0.01 atomic unit. Our calculation includes elastic scattering, direct antiproton-alpha particle annihilation and rearrangement into He(+)p(-) and ground-state positronium. Elastic scattering is calculated within the Born-Oppenheimer approximation using the potential calculated by Strasburger et al (2005 J. Phys. B: At. Mol. Opt. Phys. 38 3091). Matrix elements for rearrangement are calculated using the T-matrix in the distorted wave approximation, with the initial state represented by Hylleraas-type functions. The strong force, leading to direct annihilation, was included as a short-range boundary condition in terms of the strong-force scattering length.