Some spiders do stick to their own webs. A spider’s web is constructed of two types of silk: “anchor” and “snare” threads.
The anchor threads are used to construct the basic web; the snare threads are sticky and are used to trap insects. To get around the web, the spider uses the non sticky anchor threads.
Once in a while, though, a spider will inadvertently catch a leg or two in a snare thread. When this happens, she simply secretes an oily solvent to free herself.
Some spiders have a special claw called a scopula, at the bottom of the hind legs, that they use to slide along the sticky snare strands and help them escape from their own homespun death traps.