  The MHD equations, as a collisional fluid model that remains in local
thermodynamic equilibrium (LTE), have long been used to describe turbulence in
myriad space and astrophysical plasmas. Yet, the vast majority of these
plasmas, from the solar wind to the intracluster medium (ICM) of galaxy
clusters, are only weakly collisional at best, meaning that significant
deviations from LTE are not only possible but common. Recent studies have
demonstrated that the kinetic physics inherent to this weakly collisional
regime can fundamentally transform the evolution of such plasmas across a wide
range of scales. Here we explore the consequences of pressure anisotropy and
Larmor-scale instabilities for collisionless, $\beta \gg 1$ turbulence,
focusing on the role of a self-organizational effect known as
`magneto-immutability'. We describe this self-organization analytically through
a high-$\beta$, reduced ordering of the CGL-MHD equations, finding that it is a
robust inertial-range effect that dynamically suppresses
magnetic-field-strength fluctuations, anisotropic-pressure stresses, and
dissipation due to heat fluxes. As a result, the turbulent cascade of
Alfv\'enic fluctuations continues below the putative viscous scale to form a
robust, nearly conservative, MHD-like inertial range. These findings are
confirmed numerically via Landau-fluid CGL-MHD turbulence simulations that
employ a collisional closure to mimic the effects of microinstabilities. We
find that microinstabilities occupy a small ($\sim 5\%$) volume-filling
fraction of the plasma, even when the pressure anisotropy is driven strongly
towards its instability thresholds. We discuss these results in the context of
recent predictions for ion-versus-electron heating in low-luminosity accretion
flows and observations implying suppressed viscosity in ICM turbulence.
