That is a pressure regulator, used when the incoming water pressure is too high. It also acts as a check valve which is why a pressure accumulator is also required. It acts somewhat as a backflow preventer but may not satisfy the federal requirements for irrigation systems, if that's your issue.

Thanks! So I guess I don't have a backflow preventer. What is a ball-park figure for getting one installed?

The cone shaped thing on the left is a PRV (pressure reducing valve) and they usually have a backflow preventor built in, but not always.

IIRC backflow preventors that create closed plumbing systems in houses have been code since the mid 90's

If it does have a backflow device built in there should be a thermal expansion tank installed at the water heater cold inlet.

If you do install a backflow device you must install a thermal expansion tank.

Ask the local authority what they require. I had a standard backflow preventer that came with the house, and, when it was time for the required annual check, the inspector noticed that it was not quite the right one for our installation, and I had to replace it with an almost identical one that had one differently designed orifice.

I installed my own. I bought the correct valve online for about $125 (the same thing was $300 at the local supply house). I cut into the piping, added unions to the valve (for easy maintenance removal), and resoldered the piping; this added about another $60 in materials. It would probably take a plumber an hour to do if access is easy. I'm guessing you are looking at $500-1000, parts and labor, depending on where you live. Expansion tanks are cheap (maybe $30), but there will be more repiping for that, too.

I do have an expansion tank on the water heater. Does that mean the PRV has a backflow preventer?