I think a lot of waste management is very local, so the answer might be different for you than it is for me.
In my town, there is a box factory that gobbles up all of our paper products and turns them into new boxes. So I know that putting the cardboard in my recycling bin is worthwhile.
This is one of the Moulton barns on Mormon Row.
More info:
https://www.nps.gov/grte/learn/historyculture/mormon.htm