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tokens: for the sample of velocity-selected clusters. As in Fig. 5 , θ is the acute angle between the projected major axis of each galaxy and that of the cluster in which it resides. Cluster position angles were determined from moments of inertia of the projected galaxy distribution. The Kuiper V, Rao, and binomial statistical tests all indicate a probability that the observed distribution of angles is consistent with random BCG orientations. Figure 9. BCG alignments as a function of peculiar velocity. Here, Δ V BCG /σ is the absolute difference between the BCG’s velocity and the mean cluster velocity, and σ is the cluster velocity dispersion. 4 DISCUSSION The main findings of this paper are: A significant fraction of BCGs have spatial offsets of a few tens of kpc or more from the centroid of the X-ray light distribution that presumably traces the centre of the DM halo and reflects the cluster’s dynamical state. Compared with theoretical models, these offsets are much larger than the expected ‘wobble’ around the centre of a standard CDM halo (2 kpc versus a few tens of kpc) as in the simulations of Kim, Peter Wittman 2017 ; Harvey et al. 2017 ). However, the X-ray gas may not trace the DM halo distribution (e.g. by gas sloshing, etc.) A significant number of BCGs also have significant line-of-sight peculiar velocities relative to the cluster. These observations suggest that many BCGs are not currently at rest in the potential well of their host cluster. Despite the prevalence of BCG displacements, these galaxies still show a strong tendency to share the same orientation as the cluster in which they reside (cf. Tempel, Stoica Saar 2013 , a remarkably robust coherence of structures over scales from tens to thousands of kpc, as suggested by the simulations of Rhee et al. 2017 ). 4.1 Comparison of observations with simulations The computation of the cluster and BCG elongation axes is done as in Ragone-Figueroa et al. ( 2020 ). They are obtained from the principal axes of the ellipsoids that best describe the corresponding distribution of matter. For the purposes of this work cluster principal axes and centre of mass are obtained using DM particles within r 200 . While for obtaining BCG principal axes we use star particles inside 0.1 r 500 . Fig. 10 shows the distributions of BCG offsets in position and velocity as well as their correlations with the BCG–cluster alignment angle, in full 3D space (four left-hand panels) and in projection and radial velocity (four right-hand panels). We find in simulations galaxies with significant offsets, in projected space the mean(median) offset is ∼53 kpc(78 kpc), which is larger that the mean found in the observational data (∼15 kpc). Conversely, the mean(median) normalized 1D BCG velocity | V BCG /σ| (with respect to the cluster mean) is ∼0.09(0.07), which is lesser than in the data (0.26). The maximum Δ V /σ in these simulations is lower than that observed because of the smaller number of simulated cluster samples.