Diagnosis
The internal wires and insulation within the swivel mechanism are degrading from repeated heat exposure and twisting, losing their flexibility.
How to Fix It
Gently untangle the cord and let the iron hang by its cord overnight (unplugged) to relax the kinks. When using, make conscious full rotations in both directions to prevent overwinding in one direction.
If That Didn't Work
The stiffness is a precursor to internal wire breakage. Start planning for a replacement, as the cord assembly is not user-replaceable and will eventually fail.
Heads Up
Never forcefully yank or spin the iron by the cord. A stiff, cracked cord is a shock hazard—inspect it regularly for any exposed wiring and retire the iron if you see any.