We have performed a comprehensive numerical investigation of the scattering of antihydrogen (H¯) and positronium (Ps) atoms, at collision energies just above the threshold for the rearrangement reaction resulting in H¯+ ions. Formation of H¯+ in H¯ +Ps collisions is of vivid interest for experiments with H¯ at CERN. We treat all open channels below the threshold for the rearrangement into H¯ +Ps (which includes elastic scattering, excitation and deexcitation of positronium to and from the excited states with n=2,3, polarization and depolarization of Ps(n,l), rearrangement into H¯++e-, and formation of Ps and H¯ in the reverse reaction). Our method is a combination of the variational calculation with the coupled rearrangement channel method. The four-body solutions of the Schrödinger equation, obtained via diagonalization of the full Hamiltonian, are combined with the channel functions that satisfy the correct asymptotic boundary conditions, in a procedure that delivers the scattering matrix S. The complete set of S-matrix elements for all allowed scattering processes is calculated (both in forward and reverse directions, without assumption of the detailed balance). The scattering cross sections are obtained from the S-matrix elements calculated for the total angular momenta J=0-6, and for the positronic spin S=0, required for the formation of H¯+. The numerical accuracy of the calculation is checked using the criterion of unitarity of the S matrix. Convergence of the cross section is examined with respect to the partial wave contributions, under the constraint of conservation of total angular momentum and parity. We show that the energy behavior of the cross sections near thresholds is consistent with the Wigner laws. The cross sections for processes that compete with H¯+ formation are presented.