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We show that the fluctuations in the wounded-nucleon model of the initial stage of relativistic heavy-ion collisions, together with the natural assumption that the forward- (backward-) moving wounded nucleons emit particles preferably in the forward (backward) direction, lead to an event-by-event torqued fireball. The principal axes associated with the transverse shape are rotated in the forward region in the opposite direction than in the backward region. On the average, the standard deviation of the relative torque angle between the forward and backward rapidity regions is ~20° for the central and 10° for the midperipheral collisions. The hydrodynamic expansion of a torqued fireball leads to a torqued collective flow, yielding, in turn, torqued principal axes of the transverse-momentum distributions at different rapidities. We propose experimental measures, based on cumulants involving particles in different rapidity regions, which should allow for a quantitative determination of the effect from the data. To estimate the nonflow contributions from resonance decays we run Monte Carlo simulations with therminator, a thermal heavy-ion generator. If the event-by-event torque effect is found in the data, it will support the assumptions concerning the fluctuations in the early stage of the fireball formation, as well as the hypothesis of the asymmetric rapidity shape of the emission functions of the moving sources in the nucleus-nucleus collisions. [ABSTRACT FROM AUTHOR]