I honestly thought that these little guys were called preying mantises, rather than praying mantids. I figured that yes, they look like they are praying, but had always heard that they are fierce predators, and thus the "preying" part had stuck in my mind. It's so humbling to grow old and find how wrong you've always been!
His back looked just like a leaf, giving him maximum camouflage except, of course, when perched on a piece of wood on top of the trash barrel.
The large head of this guy turned as I moved around him with my camera. It seems that the head is on an elongated part of the thorax that looks like a neck and allows the head to turn as much as 180 degrees.
On the University of Arizona Center for Insect Science Education Outreach website, I found this information about "interesting behaviors" (I'll say!) of the praying mantid:
The adult female usually eats the male after or during mating. [A] mantid's grasping response is incredibly rapid, so that you see it before it catches the insect and when the insect is in its front legs. The motion is barely a blur if it is perceived at all.
The compound eyes are capable of seeing images and colors. The three simple eyes perhaps tell the differences between light and dark. The simple eyes are arranged in a triangle between the antennae. Compound eyes are made up of hundreds of facets constructed with two lenses. These focus the light down a light sensitive structures (rhabdome) which is connected to the optic nerve.
I checked on him some time after our photo op and found him in the same place. Given the number of predators wandering around the area (cats, dogs, birds), I gave him a ride to the far side of the fence, where he commenced to pray and prey once again.