What does a door do on its hinge?

Solution 1:

The door swings on its hinge.

To move backward and forward, especially rotating about or hanging from a fixed point.

Turns, as suggested by other answerers, sounds like another great possibility. But a quick look at the actual usage stats brings up something interesting.

Here are the top 50 collocations for "the door [v*]" from the Corpus of Contemporary American English (COCA) and the British National Corpus (BNC):

     |   COCA             |   BNC
     |                    |
 1   |   OPENED     992   |   WAS        402
 2   |   WAS        943   |   OPENED     327
 3   |   OPEN       830   |   OPEN       175
 4   |   IS         659   |   CLOSED     151
 5   |   OPENS      611   |   SHUT       103
 6   |   SHUT       588   |   IS          73
 7   |   CLOSED     573   |   SWUNG       57
 8   |   SWUNG      249   |   HAD         53
 9   |   CLOSES     164   |   SLAMMED     42
10   |   HAD        158   |   LEADING     21
11   |   SLAMMED    137   |   OPENS       21
12   |   LEADING     98   |   WOULD       20
13   |   SLAMS       92   |   BURST       19
14   |   WOULD       88   |   'S          19
15   |   SLID        80   |   LOCKED      19
16   |   'S          71   |   WILL        18
17   |   LOCKED      66   |   FLEW        17
18   |   FLEW        63   |   WERE        16
19   |   CREAKED     57   |   BEING       13
20   |   CLOSE       56   |   SLID        13
21   |   SWINGS      50   |   HAS         11
22   |   HAS         49   |   BANGED      11
23   |   SHUTS       49   |   CLICKED     11
24   |   CLICKED     46   |   MADE        10
25   |   SLAM        45   |   CLOSE        9
26   |   UNLOCKED    43   |   CRASHED      9
27   |   BURST       40   |   MUST         9
28   |   BANGED      40   |   DID          8
29   |   WEARING     39   |   SAYING       8
30   |   SAID        36   |   CREAKED      7
31   |   CRACKED     34   |   CAN          7
32   |   DID         34   |   LOOKING      7
33   |   WERE        34   |   SHUTS        7
34   |   WILL        34   |   BEGAN        6
35   |   CAME        30   |   CLOSES       6
36   |   ARE         29   |   OPENING      6
37   |   STOOD       28   |   STOOD        6
38   |   LOOKING     25   |   WENT         6
39   |   COULD       23   |   FELL         5
40   |   BEING       22   |   BROKE        5
41   |   DOES        21   |   CAME         5
42   |   GAVE        21   |   HISSED       5
43   |   BURSTS      20   |   MIGHT        5
44   |   CLOSING     20   |   SLAM         5
45   |   CRASHED     20   |   SWINGING     5
46   |   FLIES       20   |   UNLOCKED     5
47   |   REMAINED    19   |   WO           5
48   |   CAN         18   |   CLANGED      4
49   |   HIT         18   |   FACING       4
50   |   COMES       16   |   INTERRUPTED  4

As you can see, out of these swing is the top (and arguably only) verb that fits your bill, while turn is not even on the list.

Solution 2:

If turns is too bland for you, you could say that it pivots on the hinge.

Picking the "right" word depends much on your context, too.

"The door swung open" is indeed a very natural way to indicate the movement. However, if you were to, say, describe the appartus of an experiment in a scientific journal, swing might not be the best verb to use; something more "formal" like pivot may work better.

Then again, there's turn, which can be a very apt word in a metaphor, much like Wouk's metaphor quoted in Malvolio's answer to your question. In that context, I think turn works much better than swing or swivel.

Then there's flap, which you offered in your question. I think that would work best if you were describing a loose shutter in a strong wind.

The shutter turned in the wind.
The shutter swung in the wind.
The shutter swiveled in the wind.
The shutter flapped in the wind.

Which do you think sounds best?