We apply the picture proposed in a previous Letter, which relates the hyperon polarization in unpolarized hadron-hadron collisions to the left-right asymmetry in singly polarized reactions, to the production of different hyperons in reactions using different projectiles and/or targets. We discuss the different ingredients of the proposed picture in detail and present the results for hyperon polarization in the reactions such as pp, K − p, π ± p, and Σ − p collisions. We compare the results with the available data and make predictions for future experiments.