I'll start with the common setup where a free charge is placed near a wire through which current does not flow. The wire is neutral so the free charge does not feel any force upon it. When the a current is generated through the wire (meaning that electrons start moving and the atoms remain in their places, ignoring small oscillations due to thermal velocity), the explanation is that if the free charge is standing still an apparent electric force does not appear since the wire is still neutral. Only when the free charge moves with a velocity along the wire it has a force that acts on it. If we reverse the sign of the free charge's velocity, then the force changes its sign also.
Based on my understanding this picture should be asymmetrical. If the free charge starts moving with the same velocity as the electrons in the wire, then, due to length contraction/dilation, the spacing between the atoms shrink and the the one between the electrons becomes larger. This generates a charge imbalance and thus a force acts upon the free charge. However, if the free charge moves with the same velocity as the electrons but in a direction opposite to the current, then the relative velocity between the electrons and atoms in the wire won't create the same imbalance, unless their velocities are composed classically. So in this case, although the free charge has the same magnitude of its velocity, shouldn't its different direction make the spacing between the charges in the wire slightly different, thus having a different force if the free charge moves in the direction of the current or opposite to it?