An inviscid analytical theory of a slow steady liquid mass rotation during the swirl-type sloshing in a vertical circular cylindrical tank with a fairly deep depth is proposed by utilising the asymptotic steady-state wave solution by Faltinsen et al. (J. Fluid Mech., vol. 804, 2016, pp. 608–645). The tank performs a periodic horizontal motion with the forcing frequency close to the lowest natural sloshing frequency. The azimuthal mass transport (first observed in experiments by Prandtl (Z. Angew. Math. Mech., vol. 29(1/2), 1949, pp. 8–9)) is associated with the summarised effect of a vortical Eulerian-mean flow, which, as we show, is governed by the inviscid Craik–Leibovich equation, and an azimuthal non-Eulerian mean. Suggesting the mass-transport velocity tends to zero when approaching the vertical wall (supported by existing experiments) leads to a unique non-trivial solution of the Craik–Leibovich boundary problem and, thereby, gives an analytical expression for the summarised mass-transport velocity within the framework of the inviscid hydrodynamic model. The analytical solution is validated by comparing it with suitable experimental data.