  In many shear- and pressure-driven wall-bounded turbulent flows secondary
motions spontaneously develop and their interaction with the main flow alters
the overall large-scale features and transfer properties. Taylor-Couette flow,
the fluid motion developing in the gap between two concentric cylinders
rotating at different angular velocity, is not an exception, and toroidal
Taylor rolls have been observed from the early development of the flow up to
the fully turbulent regime. In this manuscript we show that under the generic
name of ``Taylor rolls'' there is a wide variety of structures that differ for
the vorticity distribution within the cores, the way they are driven and their
effects on the mean flow. We relate the rolls at high Reynolds numbers not to
centrifugal instabilities, but to a combination of shear and anti-cyclonic
rotation, showing that they are preserved in the limit of vanishing curvature
and can be better understood as a pinned cycle which shows similar
characteristics as the self-sustained process of shear flows. By analyzing the
effect of the computational domain size, we show that this pinning is not a
product of numerics, and that the position of the rolls is governed by a random
process with the space and time variations depending on domain size.
