Reds and yellows (thus oranges too) and blues were easily obtainable from many sources, so their use was quite spread, alongside the natural colours of the textiles used. Undergaments, worn to catch sweat and keep exterior clothing clean, were washed more often and usually weren't dyed, as the tint would wear off fast, unlike top layers like tunics, robes, shawls, pants... since they were more expensive. For this reason I think simpler undyed clothes should predominate on the clotheslines, just because those garments were washed more often.
Greens, pinks, purples and richer colours, alongside patterns and more intricate emboridery were used more by the higher classes due to their cost and rarity. Saturation, as poined by ItsPabs, is an important factor. Dyes need a mordant, a substance that holds the tint on the cloth. Mordants vary widely in quality an effectivenes, so more saturated, rich colours would be worn by the elite, since they were the ones able to afford them.
I think it's about the presentation and area you put these clothes as well, and I think allowing there to be room for interpretation is a good thing, imo.
Yeah, having said all that, i think this would be the easier and smarter solution. A blue cloth could be interpreted as both commoner clothes and rich, luxurious merchant textiles depending on the context where it's used. As long as the pieces aren't super bright, they could be used on both cases. It was suggested before the creation of a high class clothesline. I think this would be a cool addition. Combined with the regular would make a clear distintion between high class and commoner clothes.
Images of dyed yarns and textiles showing a range of colours moderately easy to obtain