I don't clean the rim of the bowl as often as I clean out the stem and shank, but when I feel it's time for other maintenance I will also clean the rim. I feel that the crud is cake that has gone out of bounds, and I like to look at the grain on the rim, so it it's become blackened, I use a scrap of flannel and spit to dissolve the crud. If it's too stubborn for spit I use Everclear, then apply a fresh coat of briar polish. If there's evidence of charring I will use high-grit sandpaper and micro-mesh pads to restore the wood.

Oh, and with regard to the cigar band on/off debate, I leave bands on, just in case the cap starts to unravel. sliding up the band close to the head will keep the rest of the wrapper from coming off. It has saved many a cigar.