  Magnetic helicity plays a tremendously important role when it is different
from zero on average. Most notably, it leads to the phenomenon of an inverse
cascade. Here, we consider decaying magnetohydrodynamic (MHD) turbulence as
well as some less common examples of magnetic evolution under the Hall effect
and ambipolar diffusion, as well as cases in which the magnetic field evolution
is constrained by the presence of an asymmetry in the number density of chiral
fermions, whose spin is systematically either aligned or anti-aligned with its
momentum. In all those cases, there is a new conserved quantity: the Hosking
integral. We present quantitative scaling results for the magnetic integral
scale as well as the magnetic energy density and its spectrum. We also compare
with cases were a magnetic version of the Saffman integral is initially finite.
Rotation in MHD turbulence tends to suppress nonlinearity and thereby also
inverse cascading. Finally, the role of the Hosking and magnetic Saffman
integrals in shell models of turbulence is examined.
