The statistical theory of absolute equilibrium ensembles is extended to describe ideal, three-dimensional, magnetohydrodynamic (MHD) turbulence with and without rotation, and with and without a mean magnetic field. Results from seven long-time numerical simulations of five general cases on a $32^{3}$ grid are presented. One notable result is the discovery of a new ideal invariant, the ‘parallel helicity,’ which arises when rotation and mean magnetic field vectors are aligned. Although the basic equations and statistical theory are symmetric under parity or charge reversal, the presence of invariant cross, magnetic or parallel helicity dynamically breaks this symmetry. Ideal MHD turbulence is, in general, non-ergodic due to the decomposability of the constant energy surface in phase space. Non-ergodicity can be manifested in the appearance of coherent structure as long as magnetic or parallel helicity is invariant. The fact that MHD turbulence inherently contains coherent structure in certain general cases may have important implications for dynamo theory.