  Galaxies and the dark matter halos that host them are not spherically
symmetric, yet spherical symmetry is a helpful simplifying approximation for
idealised calculations and analysis of observational data. The assumption leads
to an exact conservation of angular momentum for every particle, making the
dynamics unrealistic. But how much does that inaccuracy matter in practice for
analyses of stellar distribution functions, collisionless relaxation, or dark
matter core-creation?
  We provide a general answer to this question for a wide class of aspherical
systems; specifically, we consider distribution functions that are "maximally
stable", i.e. that do not evolve at first order when external potentials (which
arise from baryons, large scale tidal fields or infalling substructure) are
applied. We show that a spherically-symmetric analysis of such systems gives
rise to the false conclusion that the density of particles in phase space is
ergodic (a function of energy alone).
  Using this idea we are able to demonstrate that: (a) observational analyses
that falsely assume spherical symmetry are made more accurate by imposing a
strong prior preference for near-isotropic velocity dispersions in the centre
of spheroids; (b) numerical simulations that use an idealised
spherically-symmetric setup can yield misleading results and should be avoided
where possible; and (c) triaxial dark matter halos (formed in collisionless
cosmological simulations) nearly attain our maximally-stable limit, but their
evolution freezes out before reaching it.
