How do irrigation anti-siphon valves work ? Or not.?

The simplest way is a one-way valve that allows flow in one direction but not the reverse - is one of yours is gone ape - you might either want to replace it or at least clean it.

most landscape irrigation supply stores sell repair kits for most models od back-flow preventors. This is a good option but Lowe's also sells back-flow preventorrelativelyeinexpensivelyly and depending on the cost of the kit it may be more simple to just swap it out. Good Luck.