I think I've identified at least 3 different levels of stropping. Each level serves a different purpose. This list below refers to stropping on leather or some other surface with similar give. I usually use white or green compound, which seems to greatly accelerate the effect. Stropping on bare cardboard or leather doesn't seem to work very fast. With that in mind...
1. Doing 3 - 10 strokes per side. For me, this seems to remove the last little "hangs" of burr being left behind. It takes an edge from catching paper and ripping it, to clean slicing. This can be from a rather coarse stone. Same story with shaving hair: Before stropping it will catch and only shave a few hairs. After a mere 3 to 10 strokes per side it shaves much more cleanly. I guess this is like cleaning up the edge.
2. Somewhere above 10 strokes, perhaps 20 to 50 seems to be this intermediate state where the edge gets more smooth, and mostly in a bad way. At this level the shaving performance increases. The "slick" feel through paper gets better. But the ability of the edge to catch the material it's cutting is vastly reduced. I was quite surprised by this with some recent kitchen knives. Tomato slicing ability seemed quite reduced when stropped at this level. Again, shaving and paper were noticeably better.
3. At some point, perhaps after the edge has been refined with very fine abrasives (greater than fine ceramic or EF DMT), with a lot of stropping, the edge gets an extra level of polish. This is alien territory to me, as I don't have any experience with abrasives fine enough to use stropping as an additional level of polish. People talk about it here all the time.... going from some fine abrasive to compound on a strop and getting "more of a mirror polish", or just "more polish". This is either a placebo effect, or it requires very, very fine abrasives before stropping. Either way, I have yet to experience this, but have read about it many times. Long term maintenance with a leather strop is best dome on convex, Scandi, and straight razors - tools that have a lot of additional contact with the strop beyond just a thin cutting bevel. On regular V bevel there is very little room for error in pressure or angle control before negative effects start showing up.
I'm still learning here, so perhaps my observations aren't 100% correct. I'd be happy to read others opinions and experiences on this.
Thanks,
Brian.