Hard-drawn or hard-rolled products are harder (have higher y.s.) than others. They will deform less, and tend to embed less abrasives during the act of polishing. 302 and 301 are more typically supplied in sheet with 1/4-hard or higher ratings.
CPro, there are published values for s.s. in most heat transfer textbooks that are much lower for "polished" than "mill" finish. Yes, specularity plays a role in lowering emissivity, but I think the ability to clean, and keep clean (i.e. free of heavy oxidation) is also part of it. It's easier to see the dullness of oxidation build-up on a previously polished surface.