  We build on recent developments in the study of fluid turbulence [Gibbon
\textit{et al.} Nonlinearity 27, 2605 (2014)] to define suitably scaled,
order-$m$ moments, $D_m^{\pm}$, of $\omega^\pm= \omega \pm j$, where $\omega$
and $j$ are, respectively, the vorticity and current density in
three-dimensional magnetohydrodynamics (MHD). We show by mathematical analysis,
for unit magnetic Prandtl number $P_M$, how these moments can be used to
identify three possible regimes for solutions of the MHD equations; these
regimes are specified by inequalities for $D_m^{\pm}$ and $D_1^{\pm}$. We then
compare our mathematical results with those from our direct numerical
simulations (DNSs) and thus demonstrate that 3D MHD turbulence is like its
fluid-turbulence counterpart insofar as all solutions, which we have
investigated, remain in \textit{only one of these regimes}; this regime has
depleted nonlinearity. We examine the implications of our results for the
exponents $q^{\pm}$ that characterize the power-law dependences of the energy
spectra $\mathcal{E}^{\pm}(k)$ on the wave number $k$, in the inertial range of
scales. We also comment on (a) the generalization of our results to the case
$P_M \neq 1$ and (b) the relation between $D_m^{\pm}$ and the order-$m$ moments
of gradients of hydrodynamic fields, which are used in characterizing
intermittency in turbulent flows.
