I know that there are many ways to resolve a diminished seventh chord, but only a few ways to resolve an augmented sixth chord. I have done modulations from the minor tonic to the major subtonic using the spelling of vii°7 several times in my works and I thought I was maybe using a common tone diminished seventh chord. That is, until I looked more closely at how the notes move. Here's just one example of such a modulation with a Roman Numeral Analysis in both the minor key it starts in and the key it ends in.
As you can see, this modulation is one of those cases where the vii°7 vs Aug6 is not so cut and dry. On the one hand, the second chord moves similarly to your typical augmented sixth, out to an octave leading towards a cadence. On the other hand, the same chord is completely diatonic in C minor, the key it starts in, being the vii°7 of C minor. Augmented sixth chords tend to not be diatonic to either the starting key or the ending key when they are used in a modulation. So, here it is from the common tone perspective:
Common Tone Diminished Seventh
This seems to be the more natural way of viewing things given the spelling of the chord as a diminished seventh and the fact that the chord is completely diatonic to C minor, the starting key. But how does the Common Tone Diminished Seventh typically resolve? Typically, it resolves to the tonic or the dominant. Less commonly, it will resolve to either the ii or VI chords of a minor key. Either way, it typically has the same root note as the tonic chord. That certainly doesn't seem to be what is happening here. The bass is moving chromatically down from C to B♭. If a common tone diminished seventh is involved in chromatic motion, it typically moves upwards in the bass, not downwards.
From the perspective of B♭ major, the ending key, this seems to be the correct interpretation. B moves down to B♭ and A♭ moves upwards to B♭. That's the way an augmented sixth resolves, moving outwards to the octave. There's just 1 problem, I don't see an augmented sixth interval anywhere. B to G♯? That's a major sixth. D to B? That's also a major sixth. A♭ to F? Same major sixth. No augmented sixth interval anywhere. If that A were to be turned into an A natural though, there would be a clear augmented sixth interval. How can it be an augmented sixth chord if all the sixths I see are major sixths?
Either way, I end up with something that is atypical of said chord. For the common tone diminished seventh, that is the downward chromatic motion in the bass and resolution to the VII of C minor. For the augmented sixth, that is the complete lack of an augmented sixth interval. So what is it then if both the diminished seventh and augmented sixth interpretations have their flaws?