Just one question: would swapping the drive platters even work? When all is considered, a hard drive is an analog device with some electronics converting that analog signal into a digital format that a computer can handle. Any misalignment would likely distort the signal beyond use.
(Yes, I know that certain SyQuest drives were essentially removable hard drive platters. On the other hand, SyQuest drives were designed with the removable bit into consideration and they probably made some compromises to get it working.)
That said, nothing ventured nothing gained.
If you know it's something like stiction though, I would give the drive a few good twists around the rotation axis then try recovering the data immediately. If that doesn't work, I'd go onto opening the case and rotating the platters at the hub then trying it. I would not consider removing the platters since it is to intertwined with the drive heads to remove without the risk of damage.
The alignment question is a good one. In very old mechanisms, the location of the tracks is defined as in a floppy drive. Later drives use a feedback positioning method. Swapping produces no problem in the latter -- the servo will find the tracks and center the head over them.
In the case of the older open-loop drives, you may have to perform an alignment. It's easy, though, and the method is the same as in a floppy: Rotate the head-positioning stepper motor until goodness occurs.
That said, swapping platters is definitely an operation of last resort. As others have suggested here, other methods should be tried first (freezing, thwacking, swapping electronics, etc.)
