The organization of vortex clusters above the buffer layer of turbulent channels is analysed using direct numerical simulations at friction Reynolds numbers up to $\hbox{\it Re}_{\tau}\,{=}\,1900$. Especial attention is paid to a family of clusters that reach from the logarithmic layer to the near-wall region below $y^+\,{=}\,20$. These tall attached clusters are markers of structures of the turbulent fluctuating velocity that are more intense than their background. Their lengths and widths are proportional to their heights $\Delta_y$ and grow self-similarly with time after originating at different wall-normal positions in the logarithmic layer. Their influence on the outer region is measured by the variation of their volume density with $\Delta_y$. That influence depends on the vortex identification threshold, and becomes independent of the Reynolds number if the threshold is low enough. The clusters are parts of larger structures of the streamwise velocity fluctuations whose average geometry is consistent with a cone tangent to the wall along the streamwise axis. They form groups of a few members within each cone, with the larger individuals in front of the smaller ones. This behaviour is explained considering that the streamwise velocity cones are ‘wakes’ left behind by the clusters, while the clusters themselves are triggered by the wakes left by yet larger clusters in front of them. The whole process repeats self-similarly in a disorganized version of the vortex-streak regeneration cycle of the buffer layer, in which the clusters and the wakes spread linearly under the effect of the background turbulence. These results characterize for the first time the structural organization of the self-similar range of the turbulent logarithmic region.