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We conduct direct simulations of turbulent channels imposing different virtual origins for all three velocities using Robin, slip-like boundary conditions to study the effect of displacing the origins perceived by different flow components, a mechanism common to small-textured surfaces. We also explore this effect using opposition control. For riblets, Luchini et al. (1991) proposed that their effect could be reduced to the offset between the streamwise- and spanwise-velocity origins, the latter being the origin perceived by turbulence. Later results on superhydrophobic surfaces suggest that the apparent wall-normal-velocity origin could also play a role. Our results support that the relevant parameter is the offset between the origins perceived by the mean flow and by turbulence. The former is determined by the streamwise slip length, but the latter depends on both the wall-normal and spanwise slip lengths. The slip on the streamwise velocity fluctuations, in turn, has a negligible effect on the virtual origin for turbulence, at least in the drag-reduction regime. This suggests that the origin perceived by the quasi-streamwise vortices, which induce the cross velocities at the surface, is key in determining the virtual origin for turbulence, while that perceived by the near-wall streaks, which are associated with the streamwise velocity fluctuations, plays a secondary role. In this framework, the changes in turbulent quantities typically reported in the flow-control literature are shown to be only a result of the choice of origin and are absent when referenced to the origin experienced by turbulence. Other than this shift in origin, we show that turbulence remains essentially smooth-wall-like, and a simple expression can predict the virtual origin for turbulence. The effect on drag can be reproduced a priori by introducing the virtual origins into a smooth-wall eddy-viscosity framework.