Things that can be open or closed (bridges, floodgates, doors, hatches) behave as if the lever has an Open and Closed setting. Gears, on the other hand, toggle regardless of the lever state. Go figure.
To make it even more fun, if a floodgate (or whatever) is in the process of opening or closing then it will ignore all signals from a lever. If you hook a lever up to a floodgate and pull it twice in a row, then the floodgate will open and stay open, despite the lever now being in the 'closed' state. Again, gear boxes behave differently and toggle state the moment the lever is pulled so they never miss a signal.
If you want two floodgates to behave as you describe (with one open when the other is closed) then you have to build them as you described but rather than pull the lever once, pull it twice each time you want to toggle the gates.
Assuming we are starting with the lever in the "open" position, what will happen is this:
Pull 1 (lever now 'closed'): The open gate gets the close signal and starts to close. The closed gate ignores it since it is already closed.
Pull 2 (lever now 'open'): The open gate is still closing so it ignores the signal. The closed gate gets the open signal so it starts to open.
A little while later: Both gates finish their respective operations and it is now safe(er) to toggle them again without getting them both stuck in the same state.
Now if you really want to be safe, you will build a second lever anyway, because if both your floodgates wind up in sync then there is no way to put them back in opposition with just one lever.