Lithium ortho-thiophosphate (Li₃PS₄) has emerged as a promising candidate for solid-state-electrolyte batteries, thanks to its highly conductive phases, cheap components, and large electrochemical stability range. Nonetheless, the microscopic mechanisms of Li-ion transport in Li₃PS₄ are far to be fully understood, the role of PS₄ dynamics in charge transport still being controversial. We build machine learning potentials targeting state-of-the-art DFT references (PBEsol, r²SCAN, and PBE0) to tackle this problem in all known phases of Li₃PS₄ (α, β and γ), for large system sizes and timescales. We discuss the physical origin of the observed superionic behavior of Li₃PS₄: the activation of PS₄ flipping drives a structural transition to a highly conductive phase, characterized by an increase of Li-site availability and by a drastic reduction in the activation energy of Li-ion diffusion. We also rule out any paddle-wheel effects of PS₄ tetrahedra in the superionic phases–previously claimed to enhance Li-ion diffusion–due to the orders-of-magnitude difference between the rate of PS₄ flips and Li-ion hops at all temperatures below melting. This archive provides all the relevant data and input files that were used to fit the ML interatomic potentials used in this work, along with the relevant Density-Functional Theory calculations that were used for the training set construction, the validation of the ML models and the calculation of the electronic band structure of the β and γ structure. Furthermore, it provides input files of all the molecular dynamics trajectories needed to investigate Li-ion diffusion properties of Li₃PS₄ and the rotational dynamics of PS₄ tetrahedra. Finally, it provides the raw data to reproduce the figures of the manuscript associated with this archive.