Mark has his answer and the thread has a life of its own! I didn't know you could put the standard Cooper wheel cover on an S wheel, I'm not sure why I would want to, but 42 years into this hobby I have learned something new. And to me, learning something new makes highjacking a thread worth it.
Given the discussion of how to identify the different wheels, it was time for hands-on research. There's no claim that this is definitive, just what I could put my hands on yesterday afternoon in the garage.
First, going through old publications such as the Clive Trickey/Speedsport books and period catalogs like V.W. Derrington, the term "Dunlop wheels" only seems to apply to the reversed rim S wheels. Standard S rims seemed to be always referred to as just that, with no manufacture called out. That seems like the consensus already reached here. I always remembered the S wheels as having the marking "LP883", and on my search yesterday determined that that is the only marking they have. This included wheels I bought new and used, and ones that were original equipment on the cars. No manufactures mark such as RO. I believe there was an earlier post stating the Dunlop wheels have an arrow marking.
The early 850 wheels (at least the 8 or 10 of them that I have) do have the RO marking (Rubery Owen makes great sense, I had never made that connection, so learned something else here) as well as various other markings. The latter were most commonly N20 or N6Y on my collection. There are variations of the RO, ocassionally in a circle, but all of mine have it. The flange of the pictured wheel was welded to the rim by me, the weld bead is not original manufacture.
The later thick-flange 3 1/2 inch wheels from my 997 Cooper have the RO plus the part number 21A881 (verified as the BMC part number at the time) plus other markings that vary from wheel to wheel. The center is resistance welded to the rim. Did they possibly include the part number to make life easier for scrutineers?
I also came across a pair of these belt-and-braces oddballs that I can't tie to any particular car in the collection. They have the thick center, resistance welded connection plus a single rivit, and absolutely no markings. Any ideas?
It also came to light that the early 850 wheels have a much larger opening in the slot than the later wheels, something I never noticed before.
How about a set of Boy-Racer Death-Wish rims from a 1960 850 I acquired. Take a thin flange wheel and drill some pseudo-S holes in it! Edit: The elongated mounting holes are a bonus feature; Russian Roulette with five chambers loaded.
Will close with a shot for Sterling; it has big flares and the tires aren't smoking, but I was just abusing the crap out of them. It was the first time I drove this car that the owner (I now drive other people's cars) put together from his stash of good stuff. Probably 140 hp at sea level, six speed dog box, slick tires that were too wide and too tall but very sticky, and springs way too stiff. It would tire hop out of the corners (other drivers said there was daylight under both front wheels), I gave up on trying to focus my eyes, it broke the front subframe mounting bolts, and the front tires lasted about an hour. Possibly the quickest Mini I ever drove, definately the most brutal. When your only tool is a hammer, everything looks like a nail! We later sorted out the suspension and went to a legal vintage racing setup of 1293, four speed and treaded tires - still a quick car but not as entertaining.
maybe we need a thyread on Mini tire abuse.
Just a thought, when painting wheels, I used silver paint that came from any of the large manufacturers of the day, Fords usually came with silver wheels, rather than the early BMC habit of beige/fawn painted items. Dunlop steels are an excellent choice, they look great, especially with 165/70s fitted 
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