  Dynamically relaxed galaxy clusters have long played a role in galaxy cluster
studies because it is thought their properties can be reconstructed more
precisely and with less systematics. As relaxed clusters are desirable, there
exist a plethora of criteria for classifying a galaxy cluster as relaxed. In
this work, we examine $9$ commonly used observational and theoretical
morphological metrics extracted from $54,000$ Mock-X synthetic X-ray images of
galaxy clusters taken from the IllustrisTNG, BAHAMAS and MACSIS simulation
suites. We find that the simulated criteria distributions are in reasonable
agreement with the observed distributions. Many criteria distributions evolve
as a function of redshift, cluster mass, numerical resolution and subgrid
physics, limiting the effectiveness of a single relaxation threshold value. All
criteria are positively correlated with each other, however, the strength of
the correlation is sensitive to redshift, mass and numerical choices. Driven by
the intrinsic scatter inherent to all morphological metrics and the arbitrary
nature of relaxation threshold values, we find the consistency of relaxed
subsets defined by the different metrics to be relatively poor. Therefore, the
use of relaxed cluster subsets introduces significant selection effects that
are non-trivial to resolve.
