  We use a high-resolution N-body simulation to study how the influence of
large-scale structure in and around clusters causes correlated signals in
different physical probes and discuss some implications this has for
multi-physics probes of clusters. We pay particular attention to velocity
dispersions, matching galaxies to subhalos which are explicitly tracked in the
simulation. We find that not only do halos persist as subhalos when they fall
into a larger host, groups of subhalos retain their identity for long periods
within larger host halos. The highly anisotropic nature of infall into massive
clusters, and their triaxiality, translates into an anisotropic velocity
ellipsoid: line-of-sight galaxy velocity dispersions for any individual halo
show large variance depending on viewing angle. The orientation of the velocity
ellipsoid is correlated with the large-scale structure, and thus velocity
outliers correlate with outliers caused by projection in other probes. We
quantify this orientation uncertainty and give illustrative examples. Such a
large variance suggests that velocity dispersion estimators will work better in
an ensemble sense than for any individual cluster, which may inform strategies
for obtaining redshifts of cluster members. We similarly find that the ability
of substructure indicators to find kinematic substructures is highly viewing
angle dependent. While groups of subhalos which merge with a larger host halo
can retain their identity for many Gyr, they are only sporadically picked up by
substructure indicators. (Abridged)
