"Don't ask me how one ended up with reversed polarity, that is a good question for another post."
This happens because each battery or cell has a different capacity than the others it is in series with. When the lower capacity one is discharge to zero volts and there is still current draw the now dead battery/cell is reverse charged by the those that still have some capacity left.
If you draw out a series of cells, +/- +/- +/-, then assume the one in the middle has reached zero volts, you can see that the positive lead is connected to a negative point and the negative lead of the dead cell is connected to a positive point. Continued current draw will thus reverse charge the middle cell.
This is what prematurely kills many battery packs. When you continue to try to drive that last screw with your cordless drill until the chuck won't turn at all, you have just subjected most of the cells to reverse charging. Stop using the battery as soon as you notice it losing power as this is when the lowest capacity cells are going to zero volts.
This also contributes to the death of lead acid batteries when they are run dead. The lower capacity cells are reverse charged as they individually die, just as the battery you measure negative voltage on had less capacity/charge than the other. If you could measure each cell of both batteries you would probably find some positive and some negative.
Don 4107