Check out this chord:
This would be classified as a Dsus4 chord and considered to be "unstable" but I personally consider this to be a neutral chord. Here is why.
Firstly, the chord's intervals are palindromic: the intervals of a fourth, major second then another fourth are the same both directions. This doesn't really mean anything but I thought it was interesting.
Secondly, the chord is also palindromic on the spectrum of 5ths. Containing the notes D, G, and A, which are right next to each other on the spectrum: ... C [G D A] E ... . This means that the chord does not sound particularly "sharp" or "flat" (see Don Freund's Spectrum of Fifths for more information on what this means, I should probably write something about it sometime). Also interesting is that the only note doubled in the chord is the center one, which would be the center one even if reverse, further enforcing the palindromicity.
Thirdly, the chord line up symmetrically on the piano keyboard. This only applies to the Dsus4 chord (and Absus4 I guess), but makes it nice and easy to play. This point isn't really that important.
I personally find that the chord somehow sounds both neutral and dense at the same time, which gives it an interesting, mysterious yet simple effect. I use it (transposed around) in Fare Well in the middle section to this effect.
Edit: Upon reading Kyle Gann's "The Arithmetic of Listening" textbook, I found out that this chord was also the basis of ancient Greek tuning theory, with a 6:8:9:12 ratio between the frequencies. The "symmetrically on the piano keyboard" is approximately the same as saying that the chord maintains the same frequency relations when inverted, and this holds true even when purely tuned, which is part of why it had such importance. Anyway, point is it's a pretty cool chord.
But the true power of the chord isn't unlocked until you start to make polychords out of it:
This chord, because of it's neutrality, when used in polychords, brings out the quality of the interval between the chords. At least that's what I find. m2s and m3s sound darker, M3s and m6s sound more nostalgic, etc. Because the large number of notes in each polychord this can give resonance to what would otherwise be a boring interval. And because it can range from 3 to 8 pitch classes per chord, dense harmonies can seamlessly mixed with simpler ones, while maintaining a similar sound. A lot of nice properties if you ask me, which I'm presuming you do because you're reading this, which I wrote.
One more thing:
That chord, containing 24 notes, including all 12 pitch classes, is my second favorite chord. It's a 7-copy polychord of the sus4 chord, each with a bottom note an M2 higher than the top note of the last. It also can be seen as intervals of P4 seperated by M2s. This MIDI-generated recording doesn't really capture the full beauty of it when played on a real grand piano. Please go try it out if you can, you will most likely not regret it.
Disclosure: all this talk of "favorite" and "second favorite" may change in the future. This is just the case as of the time of writing.
That's about it, thank you!updated 2022-01-09