Control power output of a stack of 4 Lead Acid Battery
Well, sure you can, but that's not the same as saying you should.

Let's take your example. You've been discharging 1 and 2 while charging 3 and 4. Let's say that 1 and 2 are approximately 50% discharged, and 3 and 4 are fully charged. You're about to swap the two pairs and start charging 1 and 2.

At this point, your power requirements change, and you need 48 volts. 

So you hook all 4 batteries in series and start discharging. At some point you've discharged 3 and 4 to 50%. That means that 1 and 2 are at 0%. Any further discharge will reverse-bias 1 and 2 and cause irreparable harm. 

In other words, unless you are extremely cautious, you run the risk of destroying one or more batteries. How cautious? Depends on how badly you want to avoid blowing up your batteries. You'd need to monitor the performance of all 4 batteries and assign some sort of margin requirement to the state of charge for all batteries.

Do you really need to worry about this? I suggest you look up (if you are not familiar with it already) Murphy's Law. Words to live by.