Quoting wikipedia here but I thought this was the origination of the band...

"cigar historians credit Dutch-born cigarmaker Gustave Bock with the invention of the cigar band in the 1830s, when he ordered paper rings with his signature on them placed on every cigar intended for export to Europe.[1] In this way, an indication of quality and prestige would be lent to Bock's products, he believed."

Anyhow, I leave the band on to have something to hold on to. The idea of a sweaty last third of a cigar isn't very appealing to me.