They remind me a little of Owlflies, except that Owlflies have larger eyes, longer antennae that look more like butterfly antennae, and many of the ones around here stick their abdomens up at an angle from whatever they're perched on. You can see that this fellow's front legs are almost as long as its antennae.
The first picture shows it as it rested all day, antennae stretched forward and down, head down, as flat as it could go. Then around 8 pm, in the last of the sun, it began to move, and the second picture was taken just before it flew away. The antennae are up, the tips drooping, the head is up, it's up on its legs, and the wings aren't folded. In an enlargement of this shot, the beautiful colors in the eyes--described by someone on BugGuide.net as "fire opal" eyes, and it fits--show up.