Chords are frequency components. A seventh chord (with a "blue" seventh from the overtone series having a frequency ratio of 7/4 rather than the nominal 9/5) has the frequency relations 1 : 5/4 : 3/2 : 7/4 with a greatest common divider of 1/4, two octaves below the fundamental: if you indeed use these frequency ratios, the resulting signal has a periodicity of the note two octaves below the fundamental note. Using a "proper" seventh and/or equal-tempered tuning breaks this mathematical relation but the sound is close enough to still hint at it.
Reverting the relations yields frequency ratios 1 : 4/5 : 2/3 : 4/7 with a greatest common divisor of 1/105. Even if you use the last (lowest) note as reference, you get 7/4 : 7/5 : 7/6 : 1 with a common divisor of 1/60. Neither ratio results in a periodicity that has a reasonably straightforward relation to the original pitch.
So chords that are to some degree modeled after overtone series result "almost" in a periodic signal with a period in reasonably straightforward relation to the chord fundamental. Inverting the frequencies does not result in a similarly straightforward relation since the relation of "notes with overtones" cannot just be swapped to "notes with undertones". For starters, the various notes could not have overtones themselves for that concept to work: you'd have to work with pure sines. Those do not actually form chords as nicely as one would think as the blending of overtones is missing. And the concept of "undertones" just does not have a similar mathematical connection to periodicity as "overtones" have.
Chords and scales don't exist in a vacuum, and the tradeoff between their symbolic musical meaning and the underlying physics falls apart when inverting the frequency relations: in that case, only the symbolic meaning remains.
And that, on its own, is more arbitrary than we are used to.