It is shown how suitably scaled, order-m moments, D-m(+/-), of the Elsasser vorticity fields in three-dimensional magnetohydrodynamics (MHD) can be used to identify three possible regimes for solutions of the MHD equations with magnetic Prandtl number P-M = 1. These vorticity fields are defined by omega(+/-) = curl z(+/-) = omega +/- j, where z(+/-) are Elsasser variables, and where omega and j are, respectively, the fluid vorticity and current density. This study follows recent developments in the study of three-dimensional Navier-Stokes fluid turbulence [Gibbon et al., Nonlinearity 27, 2605 (2014)]. Our mathematical results are then compared with those from a variety of direct numerical simulations, which demonstrate that all solutions that have been investigated remain in only one of these regimes which has depleted nonlinearity. The exponents q(+/-) that characterize the inertial range power-law dependencies of the z(+/-) energy spectra, epsilon(+/-)(k), are then examined, and bounds are obtained. Comments are also made on (a) the generalization of our results to the case P-M not equal 1 and (b) the relation between D-m(+/-) and the order-m moments of gradients of magnetohydrodynamic fields, which are used to characterize intermittency in turbulent flows.