Diplacusis is a hearing disorder in which the two ears hear the same sound as different pitches. Suppose we know that someone with diplacusis has one normal ear and one ear that is slightly off. Using simple sine waves sent through headphones, it is possible to have the person adjust the pitch being sent to one ear until the sound is identical in both ears, which would tell us by how much the ears are offset (at that frequency). But is it possible to devise a test which could detect which ear is normal?
From a physics perspective: No. Relative differences can be detected as you point out, but in order to test which ear is normal, you need a reference. In the case of pitch difference, you can use one of the ears as reference for the other. In your case, the absolute reference scale is missing. And you can't get it from another person, because perception is subjective (is the pitch I perceive the same as the one you do? Why don't we have the same frequencies we don't like?). Maybe there is something objectively wrong with one of the ears or the brain tissue processing the signals - but for that, I guess you'll need to ask physicians and not the physicists. But note: "objectively wrong" here means that it is different from the majority of ears of the human population. Now you have a reference!
The underlying problem you are facing lies at the heart of experimental physics: For most experiments, you are trying to measure something on a (for all practical purposes) continuous scale: You might be measuring a frequency, a time, a distance, etc. All of this is only possible by comparison with a reference scale. In a continuum, there is no absolute scale until you fix one. This is why it is so very important to have good definitions of units: we need good absolute scales which we can compare our experiments with. Luckily, nature is in many ways discrete and has some fundamental constants that are dimensionless - and we can use that to define our scales objectively (time is defined by number of oscillations of a Caesium atom, mass will probably be defined by the number of atoms in a crystal at a specific temperature).
When the afflicted subject listens to a single tone, he will hear one pitch in one ear and another pitch in the other ear. The difference between the pitches will be clear, but for the answer to be "yes", the subject would need some other piece of information to discriminate between the two. Otherwise, they won't know if their left ear is incorrectly shifted up, or their right ear is incorrectly shifted down.
So, for the great majority of the populace who cannot listen to a single tone and tell whether it's "right" or "wrong", there's no way to tell which ear is wrong. But, those with absolute pitch CAN listen to a single identified tone and tell whether it's wrong. So, an unfortunate musician with this condition may be able to play $A$ on a piano and tell whether one ear is wrong, and which one.
As an aside, about ten years ago I had an ear infection which gave me exactly this problem. Everything sounded like dissonant chords, which (as a musician) was a serious bummer. (I knew which ear had been infected, so I knew which one was mis-hearing.)
As a software engineer, I decided to measure the condition, and so wrote a simple Macintosh application called ToneTester. It played a pitch in each ear and let you adjust the two until they seemed to match. I couldn't play the pitches simultaneously, since audio leaks from one ear to the other and you could hear the tones beat against each other and tell that they were wrong. So, my app alternated playing one tone in one ear and the other tone in the other ear. Using this I found that one of my ears would hear pitches about one and a half semitones higher than the other.
Over the following year my ears adjusted, and I once again heard clearly. I haven't updated ToneTester in a long time, and it won't run on anything later than OS X 10.6.8.