Some spiders do stick to their own webs.
A spiderweb 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.
A spider, in moving around its web, uses the nonsticky anchor threads.
Once in a while, though, it will inadvertently catch a leg or two in a snare thread.
When this happens, it will simply secrete an oily solvent to free itself.
Some spiders have special claws at the bottom of the hind legs, called scopulas, that can slide along the sticky snare strands and help them escape from their own homespun death traps.