In my experience, augmented-sixth chords are given "national" names. In C, for example, A♭–C–F♯ is an Italian augmented sixth.
Are there other traditions that teach/view this as being a chord rooted on F♯, and thus some kind of ♯iv chord?
If so, presumably the French augmented-sixth chord would be built on scale-degree 2, and the German back on ♯4. (This difference is potentially rationale against using this stacked-thirds view of these chords, but I'm still curious if any traditions do this.)
I wish I could say I learned about the augmented sixth chords originally from Piston, but I didn't. I learned the French/German/Italian names... probably from Kostka. So, what follows is from a double check of Piston's first edition harmony...
And then the chart...
Then he actually gives analysis examples using the IV and II roots...
It's interesting that Piston doesn't write #IV to show the altered root.
There are several books (I don't remember which, as many older books have different explanations) that do explain these as derived on the sharp fourth step. These explanations do emphasize the subdominant of the chords.
The important that distinguishes Augmented Sixths from other chromatic chords is the outward resolution of the augmented sixth to an octave. In C (minor or major) this is Ab-F# moving to G-G. At the same time, there is a tritone that "needs" resolution. In C the Augmented Sixths contain the C-F# tritone; as an augmented fourth; this is normally resolved outward to B-G. The German Sixth Ab-C-Eb-F# is enharmonically equivalent to Ab-C-Eb-Gb, the dominant seventh in Db. As a diminished fifth, the C-Eb resolves nicely to Db-F; the Ab moves to Db, etc. and there is the usual V7-I progression. Even in non-evenly-tempered music, F# and G# may be treated as the "same" note. Both Beethoven and Schumann made use of this equivalence (I don't remember the exact situation) by "entering" the chord as a dominant seventh and "leaving" as a German Sixth (and vice versa). This procedure affects a nice-sounding transition from C to Db or Db to C (or from one key to the key a half-step above or below.) This action is also used with other Augmented Sixths.
The French Sixth has two tritones (as does a diminished seventh). Both tritones get resolved at the same time: Ab-C-D-F# to G-C-Eb-G to G-B-D-G (or skip the middle chord). The French Sixth is enharmonically equivalent to a dominant seventh with a flat fifth (a common chromaticism). The same possibilities exist with this chord as with the German Sixth.
Some composers (in jazz particularly) resolve the Augmented Sixth by retaining one of the resolutions of the tritone(s) but not emphasizing the outward expansion of the augmented sixth. (Ab-F# resolves outward but Ab-Gb resolves inward.) This works harmonically.
Beethoven and Schumann used chord punning in several pieces (I don't remember which.)
This particular "interval pattern" of any of the Augmented Sixth resolutions is sometimes moved to another scale note. It doesn't get much discussion but a few books discuss the procedure. It's another method of transposition.
Personally, I like to treat the Augmented Sixths as having their bass note as the "root" and not worry about names. The Stradella accordion bass pattern does have an Italian Sixth on the nominal bass note which allows all the bass chords to have the same number of notes and create Augmented Sixths if needed.
Some references: https://www.proquest.com/docview/304122215?fromopenview=true&pq-origsite=gscholar
https://www.proquest.com/docview/1652862890/887FB028EC4C4252PQ/1?accountid=147094
