A hexagon is a boundary shape with the least energy. What you are really probably seeing is a circle that has pushed outwards against another set of forces and reached equilibrium (differing temperatures, gasses, who knows what). I have been playing with voronoi diagrams and diffusion limited aggregation recently , and it is amazing some of the structures that will emerge when a system reaches equilibrium. It is possible that certain forces have aligned with their peak energies in the form of a regular polyhedra, which happens all over nature on large and small scales.
So the above is a factual statement of reality. A wild conjecture is this metalworking effect might be a physical manifestation of what the OP is talking theoretically about. A further (pair of?) wild conjecture is either my example or the OPs example could have something to do with saturn's hexagon. I would not be completely surprised if there is a numerological oscillation set up where 1/6th (or 1/3rd) of a planetary rotation is exactly long enough for prevailing winds or a sonic boom or something like that to cross the diameter of the hexagon. Or we're seeing a weird phase change trick as "something" evaporates and condenses in response to a rolling atmospheric high pressure zone.
There's also a circle packing argument where at a deeper atmospheric layer than we can see, there's six rotating vortexes which just fit within the hex.
All I can say is I'm thankful its a stable 6-agon instead of a stable 5-agon or we'd never hear the end of it from the nutcases.
If it were science fiction there would be a giant hexagonal shaped structure under the clouds on the pole. Waiting ...
Don't take it off! Saturn will fall off and go wobbling around the Solar System.
Wow. Imagine if it were a square on Saturn. I think that would provide more fodder for sci fi writers than the hexagon.
The link mentions "Saturn's South Pole … with its rotating vortex" with a link to another post from the website with a picture of the vortex, but the description is that of "the north pole of Saturn… [a] vortex of strange and complex swirling clouds" encompassed in a hexagon.
Do we have pictures of the south pole?
"Convective Bénard cells tend to approximate regular right hexagonal prisms, particularly in the absence of turbulence"