If it's a MK 1 S box it will most probably have 12FA xxx stamped on a pad just below the starter motor.
12FA is an English engine number prefix used for, I think for MG 1100 and other non 'S' BMC 1275 twin carb engined cars. (Not 100% certain about this) My OZ Mini workshop manual gives UK built 1275 Cooper 'S' cars as using 12FA engine prefix numbers but this is incorrect as all Cooper cars, 997, 998, 970, 1071 and 1275 including OZ built 'S' cars used 9F as an engine prefix. This is the exception that proves the rule that BMC used 8, 9, 10, 12 etc to denote the engine capacity. Mini Coopers are different! The only ones that did not follow this were the UK MK 3 Cooper 'S' which I'm pretty certain used the 12H prefix, as well as the OZ Clubman GT which used 1200 for the Cooper 'S' engined cars, apart from the first 100 or so cars which used the 9FXEY prefix. (OZ built 998 cooper cars may have used an OZ engine number. 9Y??????)
By changing the diff housing during manufacture it could be either a remote box or a magic wand box but the bit that holds the gear set is the same or commonised. Prior to this there were different castings depending on the application.
Hope this explains it.
RonR