Good day all,
This is a very interesting thread, as well as the pictures on the JPL / NASA sites.
The large "hurricanes" on Saturn are essentially "polar vortexes", and like the (much weaker) ones that cause polar easterlies on earth (low pressure aloft over high pressure at surface), they are "locked" at the poles.
Often, planets with a gas atmosphere, even the thick one on Venus, rotate faster than the planet rotates (like a jet stream, but larger) ... For example, Venus rotates every 225 days or so (don't quote me on this), but it's upper atmosphere rotates every 4 days (this is called super-rotation) and is also found on Jupiter, which - despite it's immense size - rotates once in an impressive 10 hours. Saturn's is just under 11.
With the Gas Giant planets (Jupiter, Saturn, etc), the immense speed of rotation at the equator is at it's "virtual surface" (cloud deck / 1000 MB height). These planets have no solid "surface", just increasing density (until liquid density is achieved) as one procedes deeper into the atmosphere. The centrifugal effect of this rotation is so great, it makes Saturn visually "flattened" at it's poles.
On a "solid" planet, like earth, the rotational speed decreases from the equator when one moves towards the poles as a Cosine function of the lattitude angle (half the speed of at the equator at 60 deg north / south lattitudes).
With a gas giant, there is a high probability that not all layers of the atmosphere rotate at the same rates (such as with super-rotation). The massive polar vortex on Saturn can be attributed to the strong winds adjacent to lighter winds (and less rotation) at the poles, probably above slower moving (or faster moving) winds.
The "coriolis" force, as it is on earth as when an air parcel changes lattitude and is affected by the rotation speed, will be MANY times stronger on a rapidly-rotating gas giant. Jupiter's big red spot is proof of this. Similar storms appear on Saturn but are shrowded (in contrast) by the haze.
Convergence, mixing, and convection attrinute the the small (small being thousands of miles across by Earth standards) cloud formations within the vortex. The "hexagon" shape can be attributed to small waves / sub-vortices within the eye wall of the vortex (like those found in polygonal eyes of hurricanes here on earth).
Everything on these planets is BIGGER, including the lightning (as Saul mentioned). Yes, there will be tornadoes and such ... However, they cannot be true tornadoes as there is no "surface" to touch down on - So I guess I'll call them funnels (just big ones).
Thunderstorms on such planets will also behave like their much smaller counterparts similar to elevated storms we see in the western US on Earth. There is no surface for parcels entering them to work with. No gust fronts either, unless an iscentropic / density surface is encountered to falling air (or in this case, gas) parcels. This should explain the lack of outflow boundaries / squall lines (no cold pool to spread out along the "ground").
Lightning on such planetary scales would be hundreds or even tens of thousands of miles long, and entirely cloud-to-cloud as "dendrite" / anvil crawlers. Anything struck by such lightning will face certain vaporization.
Another consideration is the composition of what is actually doing the "phase change" to release latent heat (that is being transported from lower layers of the atmosphere). Water vapor provides the transfer on earth to drive storms ... On the Gas Giants, Ammonium ice, Methane, Hydrogen, etc play these roles (along with a trace of water).
Intense radiation influx is also another consideration. Jupiters magnetic field is so strong that is focuses charged solar particles - even at such a great distance from the sun - So much that an up-protected astronaut will be killed from radiation sickness in minutes. The correlation of solar radiation and lightning has a mixed debate here on earth, let alone it affecting a Jovian / Gas Giant planet - Just another thing to consider.
My 2 cents worth...