  We investigate long-time properties of three-dimensional MHD turbulence in
the absence of forcing and examine in particular the role played by the
quadratic invariants of the system and by the symmetries of the initial
configurations. We observe that, when sufficient accuracy is used, initial
conditions with a high degree of symmetries, as in the absence of helicity, do
not travel through parameter space over time whereas by perturbing these
solutions either explicitly or implicitly using for example single precision
for long times, the flows depart from their original behavior and can become
either strongly helical, or have a strong alignment between the velocity and
the magnetic field. When the symmetries are broken, the flows evolve towards
different end states, as predicted by statistical arguments for non-dissipative
systems with the addition of an energy minimization principle, as already
analyzed in \cite{stribling_90} for random initial conditions using a moderate
number of Fourier modes. Furthermore, the alignment properties of these flows,
between velocity, vorticity, magnetic potential, induction and current,
correspond to the dominance of two main regimes, one helically dominated and
one in quasi-equipartition of kinetic and magnetic energy. We also contrast the
scaling of the ratio of magnetic energy to kinetic energy as a function of
wavenumber to the ratio of eddy turn-over time to Alfv\'en time as a function
of wavenumber. We find that the former ratio is constant with an approximate
equipartition for scales smaller than the largest scale of the flow whereas the
ratio of time scales increases with increasing wavenumber.
