What is the purpose of these 1-foot-square vents in the ceiling, which have a loose thin sheet of some white board on top of them? These are in most rooms in a 50-year-old single-story residence in Arizona. The house was retrofitted at some point with central air. ETA: These open directly into the attic and are not tied into the AC duct system.
Did the house once have a whole-house fan? You may see a rectangular patch on a gable end of the attic where the fan used to be. If these vents opened directly into the attic space, they would have helped to ventilate each room via the whole-house fan. This would have worked especially well at night by opening windows to draw in the cooler outside air.
Did the house once have a swamp cooler? They put those vents in as a release for the air. When the swamp cooler is running if the humid air has nowhere to escape the house expands and swells like a ballon, causing cracks to the structure. I would recommend closing them off if the swamp cooler is no longer used our installed. It just lets all that AC air out into the attic every time you open a door in the house. Makes for a very expensive electric bill in AZ!