I was just running through some augmented chords using the circle of fifths, and was continually losing my place. After a bit of thinking, it dawned on me that this was because and augmented chord has the tones for two other augmented chords.

For example, if you are in C, the notes are C E G#, inversion of this are E G# C, and
G# C E , both of which are also augmented.

So my question is how does one use this piece of knowledge? Are there interesting reharmonization that one can use this for?
(im going to think about it myself too, i'm just seeing if you guys can point me in the right direction first)

I have nothing witty to say here at the moment

Expect a change soon
Well, you could try the augmented as a dominant substitution, and then of course interpreting a reharmonization as another key's dominant could give you an opportunity to modulate. But that's not the only thing you can do; experimentation will no doubt give you some interesting results.

I find the whole tone scale (which contains two augmented chords, and therefore [by reharmonization] 6 augmented chords) to be fun in general, so for moments of tonal ambiguity in a piece, you could move to an augmented chord, and move around in the corresponding whole tone scale for a bit, before either returning to the original key or modulating somewhere else.
(Slightly outdated) Electronic and classical compositions by m'self: Check 'em out
Well, there are four different augmented chords.

Each of those augmented chords can suggest 3 different melodic minor keys (for a total of 12) (can you smell the pivot possibilities?).

Lower any pitch in each augmented chord by a semitone and it becomes one of 3 (again for 12 total) major triads.

Similarly, raise a pitch to get minor triads.

Be careful with it, though; the strong symmettry and structure found in both wholetone and diminished harmony means it can sound bland very fast if it's overdone.