Trying to understand Stravinsky’s compositional process is probably impossible, or, at least, improbable. The "Neo-classicism" of Stravinsky is, in my opinion, a musical parallel to the art world's explorations of Dada and Surrealism. If you look at Dali's bent watches and try to tell the time you are attempting to do the same thing as making sense of Stravinsky's triads (non triads). You are correct that the harmonic material in Pulcinella ranges everywhere from completely tonal to completely bent out of shape—kind of like walking past fun house mirrors. Remember, that you are trying to understand the mind of a great composer who was infinitely playful, comical, and a major troublemaker. This will not solve your technical problem but I hope it gives you some perspective as you try to sort out your data. By the time Pulcinella came along the diatonic tonal system was dead and buried. Stravinsky was using exhumed parts of this dead system in the same way that Dr. Frankenstein was trying to create new life from old parts. Got the picture? What made Stravinsky so great was the fact that no matter what he stole (and he stole a lot of different stuff throughout his long career) he processed it in his own unique way and it came out sounding like Stravinsky. You may come up with a lot of data and exquisite formulas but pinning down Stravinsky is much like herding cats. Late in his career, when Stravinsky finally adopted many of the serial techniques invented by Arnold Schoenberg he used them in a way that sounded nothing like his Viennese contemporary. His ballet Agon is as fresh and bright as his earlier Petrushka, only the harmonic materials of tonality have been replaced with the vertical byproducts of tone rows. What is amazing about Stravinsky was his ability to reinvent himself with great regularity much like his contemporary, Picasso, an equally dangerous provocateur who used distortion to great effect.
Jablonsky, Stephen. “Stravinsky: A Short Take.” More Than You Wanted To Know About Music, 2018, p. 105. CUNY Academic Works, academicworks.cuny.edu/cc_oers/41. CC BY-NC-ND 4.0.