From "How to Make and Use the Storage Battery: Its History, Theory, Maintence and ... " by Percy B. Warwick (1903) pages 81-2, see attached. Plates need to be exchanged to reverse the reaction so that rather than charging discharging is achieved. The energy is stored chemically in the electrolyte.
I looked at the two patents by Friedrich Marx, a Swiss one with the number CH2178A and one from the US number 440,175. The one "explains" the mechanism and the other explains why it is not really necessary to swap electrodes (metal electrodes will decay in HCl solutions) if one follows a specific design.
I was curious about the mechanism which clearly - by the color changes - involves going from Fe2+ to Fe3+ during charging and reverse for discharging. This makes me come up with two tentative reaction equations, one for charging
If this is right, per cycle a little bit of water is spent on gaseous hydrogen and oxygen that - by the way - are not produced simultaneously here.
What needs to be done is check whether these reactions are feasible. Maybe I will have some time to look into this; the required thermodynamic data is available I think.