Trust me Vic, My wife has a MK2 Si 1.4 iVtec a couple of years younger than your Jazz, the clutch is the same - and if I shine a torch on the pulley 2 below alternator I can see the centre of pulley stop and rotate as the aircon cycles. It is the only way to stop compressor running.... Some car makers have a piston type variable displacement compressor where it can in theory stay connected to the pulley and the displacement varied from pretty much zero to maximum. Honda have a fixed displacement scroll type compressor and the only way to stop it pumping gas at full capacity is to disconnect it from the pulley, that is when you hear the fans start and stop, the clutch cuts in and out at same time. You need a torch to see pulley even in daylight as it is so deep down, partly hidden by a chassis member. If your 2010 jazz has any other system it will indeed be an oddball.
Many parts sellers call the 2002 --> 2008 Jazz the 'MK2' ( I saw Mk2 in the first link I posted, did not look at dates below, I would have if I had wanted to buy it ). This seller ( link below ) says suitable for 2010 'Jazz MK3' ( which we on clubjazz call the MK2 because this forum does not cover original 1982 --> 1986 Jazz ).
https://en.wikipedia.org/wiki/Honda_Jazz, which bears more than a passing resemblance to the Austin Metro
https://en.wikipedia.org/wiki/Austin_Metro which Austin Rover released in early 1980's, at the time when Honda and Austin Rover had a technical tie-up.
You can clearly see the front view of clutch showing the triangular piece and the 3 radial springs to open the clutch when it gets de-energised, that triangular piece is the bit that rotates and stops..
https://www.ebay.co.uk/itm/124579639417?epid=6039437625&hash=item1d01867079%3Ag%3AijQAAOSwL0pgLooh&fits=Car+Make%3AHonda%7CCars+Year%3A2010%7CModel%3AJazz
