Yes, there is reason to believe that positive ions are going in both directions, and because in both cases the motion is non-Newtonian, in both cases the motivation has to be electromagnetic. So how could that be?
I'm contending that there is an electric field between the Sun, which has a net negative charge, and the heliosphere, which has a net positive charge. That will get positive ions moving toward the Sun. In the case of highly ionized iron atoms spewed out in CMEs, the acceleration back into the Sun can be relativistic.
But we also see lighter elements (especially hydrogen) streaming out from the Sun in the solar wind, at up to 800 km/s. That's not really what you would call a relativistic velocity, but it's way past anything predictable on the basis of the thermal velocity (even at millions of degrees), so this is clearly non-Newtonian as well. So I'm contending that these positive ions are motivated away from the Sun by electron drag. Given the flow of electrons away from the Sun, some of the positive ions will get drug along with the electron stream. Once accelerated in that direction, the ions become candidates for electron uptake. As neutral atoms, the electric field no longer affects them, but they still might be hit by a free electron every now and again, and further accelerated. Hence in an electric field, as a general rule, positive ions go one way and electrons go the other, but the further you get from the cathode, the more chance there is of finding counter-streaming atomic nuclei.