I've enjoyed reading the various recollections in this thread, many of which I share too.
The practice of collecting a recycling unwanted textiles began in the Heavy Woollen district of West Yorkshire, of which the largest town is Dewsbury.
As Ring Warrior mentioned, the rags were indeed converted into 'shoddy' and 'mungo', from which an inferior woollen material was generated in a process invented by Benjamin Law in 1813. By 1860, the neighbouring town of Batley was producing around seven thousand tons of shoddy per year, an industry that consisted at that time of around 80 firms employing a total of 550 people to sort the rags for the shoddy manufacturers.

Although this mill has now been converted into dwellings, it is probably a listed building and its former association with the shoddy trade is still evident in the sign on the wall.
Wool is graded into various categories, the main four in descending order being 'tops', 'noils' 'shoddy' and 'mungo'. Each category is basically the leftovers from having combed out the previous one. The 'tops' would be used for clothing and quilts, the 'noils' were more likely to be used for carpets and cheaper clothing, shoddy would probably end up as felt and mungo as soundproofing material for car engines.
Re-manufactured wool, from ripping up old woollen cloth, would begin as shoddy and the leftovers from that would be mungo.

This is multicolour cotton and wool shoddy.