Fascinating and illuminating! I went back and examined all my taxpaid liquor stamps to see which ones used the purple backing. All those with the purple strip were series 1875 or later. And, as you point out, the purple strip always runs behind the serial number. Since my collection is nearly all remainders we have to conjecture what became of used taxpaids. It sounds like they were glued to a cask or barrel, then the part of the stamp with the serial number was torn off, still leaving the purple strip stuck to the barrel. It's no wonder so few cancelled copies exist!
In my collection, here are the stamps which have the purple backers:
Rectified Spirits - 30 gal., Series 1875
Wholesale Liquor Dealers - 10 gal., Series 1875
Wholesale Liquor Dealers - 5 gal., Series 1876
Distilled Spirits for Exportation - Series 1878
Imported Spirits - 5 gal., Series 1879
And interestingly here are a few that did NOT have purple backers:
Distillery Warehouse series 1875 and 1878
Special Bonded Warehouse stamp for Grape Brandy, series 1877 and 1878
Most interesting was the crossover year of 1875. The 30 gallon Rectified Spirits stamp has the backer -- AND it has TWO serial numbers on the stamp itself (not including the one on the stub). The backer runs under the one on the right side. Next I have a 10 gallon stamp from the same series, with only one serial number in the center and no purple strip. And lastly, to confuse things, a 5 gallon stamp also from the same series with TWO serial numbers and NO strip.
I think for the 1875 series some experimenting was still going on during the printing process and when the purple strip was attached. It appears to me that the strip was attached after printing and probably after numbering. Perhaps the Treasury Department at first thought it might be a good idea to leave one serial number on the cask and remove the other, then later decided that wasn't necessary. I welcome further theories.