The vents are hard to explain unless you have one in front of you. The whole unit simply slides out if you pull it.
At the top (where you normally see) are two parts as you know and if they swivel independently then the cross member which links them has come loose.....there are fragile little pips that it locates into.
Here are the two parts from beneath showing the linking bar removed from one of them - yours is probably like this...
Here's the top view of the two parts of the vent lid...
The two lid parts attach into the casing via two pips - these are the pivot points and are easily broken....you may find that this is the root cause of your problem. You can make and glue in another pip yourself. In fact if you look closely at the above pics you'll see that some of the pips are missing from mine and i've glue in some new ones after drilling out the old stumps.....if the pivot pips are missing then the lids move incorrectly and come away from the linking bar.
If you remove both lid flaps you'll see this....
This is the actual flap that opens and closes the air vent. The lever attached to it locates into one of the two lid flaps....here it is disconnected and one lid flap removed....
The 'eyelet' in the centre needs to be pushed into the slot in the lid flap. Then the flap joining bar attaches the two lid flaps together.
This shows it nearly in place...
Sorry but it's hard to explain and to photo.
Take your vent out and you'll probably find that at least one of the pivot pips on the lid flaps has broken and that the bar that joins the two lid flaps together has come off one of them. Fix the pivot pip and then attach the adjoining bar.
Take it apart and all will be revealed. Failing that - go to a breakers yard and see if you can get one or tey one of the ebay auctions that are braking 156s. Fixing them is very fiddly even if you take it out of the car.