Using direct numerical simulations with grids of up to 512(3) points, we investigate long-time properties of three-dimensional magnetohydrodynamic turbulence in the absence of forcing and examine in particular the roles played by the quadratic invariants of the system and 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 either become 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 already predicted by statistical arguments for nondissipative systems with the addition of an energy minimization principle. Increasing the Reynolds number by an order of magnitude when using grids of 64(3)-512(3) points does not alter these conclusions. 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 quasiequipartition of kinetic and magnetic energies. We also contrast the scaling of the ratio of magnetic energy to kinetic energy as a function of wave number to the ratio of eddy turnover time to Alfvén time as a function of wave number. 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 wave number.