The use of keys has always played an important role for every composers.
Since the first Viennese School the use of a specific tonality was able to characterize a musical composition on a metalevel. Important examples of that are the Beethoven’s Symphony No. 3 Op. 55 “Eroica” in E-flat major or the Mozart’s  Requiem Mass K. 626 in D minor. 
Igor Stravinsky had a great knowledge of the music of those composers and the reflection of that is clear in his opera-oratorio Oedipus Rex.
Well, Stravinky’s music does not actually conform to the “common practice period” of Beethoven and Mozart… and because of that his vision of tonality is completely different: since it is based not only on a single key but in the “polar attraction of sound, of an interval or even a complex of tones” (Stravinsky, “Poetics of Music”).

Central element of Stravinsky’s music is the interval of a minor third. This interval in fact is present right on the beginning of the composition, that opens with a B-flat minor chord but, the continuous oscillation from B-flat to D-flat, creates the minor third pole that is recurring into the entire work. The composer’s knowledge about the use of tonality during the classical era becomes obvious in the character of Oedipus: Stravinsky decided to assign him the E-flat pole, which is the same of Beethoven’s “Eroica” and “Emperor piano concerto”.

From now on E-flat will be the pole of the “Heroic Oedipus”. The B-flat beginning pole, present during the supplication of the chorus, is related to the Oedipus’ E-flat, creating a Dominant-Tonic relationship.
Another Tonic-Dominant relationship is created by the “C” pole in which the Oedipus music has a cadence (and also represents the Gods’ key) and the “G” pole that is the pole of the “revealing of the truth”.

Moving on into the work we can see how Stravinsky decided to characterize not only characters but also the events that they are going through with the use of keys. When Tiresias responds to the Oedipus’ accusation of coveting the throne with the word “Miserande” Stravinsky decided to use an A minor triad, chord that is a triton apart from the heroic E-flat. This A minor pole will be associated with another sad announcement later in to the piece.
Another important pole is the “D”, this pole concludes the Tiresias’ aria where he reveals who is the murderer of the King. This pole is one of the most important of the entire work because it represents the concept of “truth”. Right after this chord Stravinsky decided to use the “D” note as a leading to tone to the Oedipus heroic E-flat. Also the pole here is oscillating between E-flat and C demonstrating the two aspects of Oedipus personality.

Another appearance of the “D” pole is during the Jocasta’s air during the second act. In the middle section of this aria the music is oscillating between D and F. The “F” can be seen as an attempt of searching the truth but the presence of the “D” affirms that what she is saying is the truth. Another appearance of the minor third relationship is during the duet between Oedipus and Jocasta, here the pole is C minor, a minor third from the heroic E-flat. This pole reflects the “dark side” of the heroic Oedipus.

Another important role here is played by the tonal pole “G”, in fact this is the pole of the messenger’ last proclamation (the first one was in the A minor pole, recalling the Tiresias’ “miserande”) in which the announces the death of Jocasta. After this proclamation the chorus sings the same music that opened the opera. The difference here is that the B-flat/D-flat relationship is replaced by G-flat/B-flat meaning the acceptance of the truth.

As we saw Stravinsky uses pole in a small and a large dimension. He used sometimes only one note or one chord (the A minor “sadness” chord on Tiresias’ “Miserande”) to underline a particular action or feeling or an entire key/pole (the heroic Oedipus E-flat). One recurring element is the interval of a third, that relates all keys together (B-flat/D-flat, B/D, ecc).