But regarding a gas giant world, do different parts rotate at
different intervals?


Length of Saturn's Day Updated Based on Mystery Signal

http://www.space.com/scienceastronomy/060503_saturn_day.html

How long is a day on Saturn? According to a new study, it's 10 hours, 47 minutes
and 6 seconds—give or take 40 seconds.

That does seem to be a problem in calculating the "day" of a gas giant world; there's a certain degree of ambiguity in picking out the part which is the official "clock" of the planet. (Similarly, Earth's core rotates slightly faster than the rest of the planet -- but in that case, we just pick the rotation rate of the rigid surface. In the case of the giant planets, the only benchmark they can pick is the rate at which the magnetic dynamo, and the field which it generates, rotates, since they can measure the effect which this has on the fluctuations in the planet's magnetospheric radio-wave emissions.)

I'll have to review my material on this, but it appears that they had particular trouble locating a portion of Saturn's internal magnetic dynamo structure that didn't change its rotation rate modestly over a period of a few years -- or even a few months. There has been some speculation that this -- combined with the fact that Saturn's magnetic axis is lined up virtually perfectly with its rotational axis, unlike any other known world or star -- is due to the fact that the magnetic-field generating process for Saturn is entirely and radically different from that for Jupiter. But then, planetary magnetic fields -- including that of our own planet -- remain one of the biggest mysteries in modern geology, and "Juno's" extremely detailed studies of the structure of Jupiter's magnetic field will be very important in providing new insight even on our own planet's field.