You might be interested in the writings of Schenker. Although he was thinking of tonal systems, the principle still applies: there can be a "force" on the foreground, the middle ground, and the background, each of which operate on a different timescale.
Take for example a traditional 64-53 half cadence at the secondary dominant in C (G). The first chord has the structure of a tonic in G, but the G note is not at rest; it is pulled down (at the foreground level) by the suspension. At the same time, it is at rest (at the middle level) because it is the tonic note in the current key. In the background, however, it still exists in a piece in the key of C, and is operated on (weakly) by the background forces that insist that it isn't the tonic, it's the dominant. Thus the note is pulled on by three different forces all at the same time. That's Schenkerian analysis in a nutshell.
As for Schönberg's writings, he seems to be claiming that forces exist when a note belongs to an overtone series, and because C belongs to a series of overtones that originates in F, its relationship to F is just as strong as its relationship to G (which is obviously in C's overtone series). This is typical Schönbergian thinking-- he seems to have a strong orientation toward symmetry, which gives some insight into his later (atonal) works. In my humble opinion though he has made an unjustified logical leap, as he has not presented a theoretical framework in support of such symmetry, and is conflating Klang with Dreiklang (reference). A C Klang implies a C Dreiklang, in which an F Klang simply doesn't exist-- you'd have to go all the way up to the 21st partial to even come close. It is a shame, though, since an eye toward symmetry quickly explains a minor triad in the context of Naturklang (since a minor triad is a major triad that is upside down), and the very existence of the minor triad was a pesky problem during Schönberg's time-- it doesn't fit the Naturklang ("Great Major Triad In the Sky") approach to harmony theory at all. The present explanation on page 23 of his book doesn't address this, nor can it, since the foundation of his argument is the overtone series, which doesn't have a minor third until maybe the 45th partial.
In an attempt to be charitable to Schönberg (since I love his Verklarte Nacht he must not be a total idiot), it is worth mentioning that music theorists do often talk about the "plagal" and "authentic" side of a key. The plagal side is downward, toward F and the flat side of the circle of fifth, while the authentic side is upward toward G and the sharp side (corresponding to the plagal and authentic modes that make up Gregorian modes. A musical work can exploit these two sides as it explores different key areas, and taking note of this can offer insight into the relative stability of a key within a larger context-- the farther up or down you go on either side, the farther you get from a stable center. This is a bit different from saying that the key is "pulled" in either of those directions, though, and non-sequitur to the notion that it is pulled in both simultaneously.
You might also be interested in the writings of Victor Zuckerkandl as well, especially Sounds and Symbol, in which he states:
A system in which the whole is present and operative in each individual locus, in which each individual locus knows, so to speak, its position in the whole, its relation to a center, must be called a dynamic system. The dynamic qualities of a tone can only be understood as manifestations of an orderly action of forces within a given system. The tones of our tonal system are events in a dynamic field, and each tone, as it sounds, gives expression to the exact constellation of force present at the point in the field at which the tone is situated. Musical tones are conveyors of forces. Hearing music means hearing an action of forces.
Victorkandl's work (about three decades after Harmonielehre) seems to pick up where Schönerg leaves off, and goes a step further: there aren't just two forces, there is a whole "constellation" of them, and the musical context forms a dynamic system. His work goes a step further along the fundamentalist view of the Naturklang, notably diverging from other contemporaries such as Carl Seashore who begin to place more and more emphasis on psychology and pattern-matching approaches to music. Indeed, the last section of the book, entitled simply "Triad," can justifiably be seen as further rumination on the concept of the Naturklang and the fundamental nature of harmonic space as separate and distinct from physical space, with its own structure and its own laws. And in the end, those laws seem to defy analysis, leaving authors like Schönberg and Zukerkandl grasping at straws (and, apparently, writing tome upon tome) trying to explain them.