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When putting my Christmas wrap, ribbon, bags, etc. away I noticed the rolls wouldn't stay rolled up without the aid of tape (which ruins the paper). I'd just removed a pair of knee high nylons, with a run in one of them. After washing the nylons, I cut across the legs in strips, slid them on the rolls of paper and Voila! Neat rolls and recycling too!
The best way to store wrapping paper is to unroll it from the cardboard tube and then re-roll it and store it inside the tube.
Source: Been doing it for years