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I had been told that the AC compressor briefly cycles every so often when off. It keeps everything circulating to prevent issues.
NO. Not on any 2010 and older Super Duty models. I don't think 2011 and later do it either.
There are a number of vehicles that turn on compressor clutch during starter operation so the compressor turns 2-3 revs to clear the compressor of liquid refrigerant. This is done so when it does come on a few seconds later, when engine speed is higher, there isn't a stress on compressor from a near hydrolock and a resulting thump noise that can be heard in the cabin. SD models can have that thump in certain conditions and has often been wrongly claimed to be a servo opening or closing a door in HVAC case.
CASS can be implemented only on PCM-controlled compressors (which is not what the OP has or what has been under discussion). This sounds like what you're thinking of.
The following is an excerpt from the 2008 f-SD WSM which I just happened to have open at the moment. Other models are probably similar.
Compressor Anti-Slugging Strategy Liquid refrigerant may accumulate in the A/C compressor under certain conditions. To alleviate damage to the A/C compressor, compressor anti-slugging strategy (CASS) is utilized. CASS is initiated only under specific conditions:
The ignition is off for more than 8 hours
The ambient temperature is above -4°C (25°F)
Battery voltage is above 9.5 volts during engine cranking
When these conditions are present, the PCM will activate the A/C control relay prior to cranking of the engine. The A/C control relay engages the A/C compressor for approximately 5 A/C compressor revolutions or a maximum of 2 seconds (depending upon vehicle application), allowing the liquid refrigerant to be pushed from the A/C compressor. CASS is initiated by the PCM regardless of the function selector switch position or the electronic automatic temperature control (EATC) system settings.