This is the first webspinner I've seen! Kinda cool. Going by this comment on BugGuide and pics elsewhere on the web, I'm guessing this is Haploembia solieri, mostly due to the faint spots on the back of the head. Apparently it's not native to CA. Boo.
Webspinners or embiids (order Embioptera) are a small group of mostly tropical and subtropical insects, classified under the subclass Pterygota. The order has also been referred to as Embiodea or Embiidina The name Embioptera (lively wings) comes from Greek, embio meaning lively and ptera meaning wings, and refers to the fluttery movement of wings that was observed in the first male specimen described.
