Here is a discussion of Coltrane's approach to harmony in this and other pieces. The technique involves third-related harmonies which, since the relationships are symmetrical, avoids giving a sense of harmonic direction. The Wikipedia article on the Coltrane changes first describes the progression in "Giant Steps" as C, Eb7, Ab, B7, E, G7, C or, in terms of harmonic analysis: I, V of bVI, bVI, V of III, III, V7, I. Simplified a bit to show the movement down by major 3rds: I bVI, III (V7) I.
Later on in the article the chords given there are B maj 7, D7, G maj 7. Bb7, Eb maj 7, A min 7, D7 which is how they appear in the chart for "Giant Steps". The tenor sax is a transposing instrument in Bb, meaning that a written C sounds as a Bb, so if you want a C, you write a D. But that still doesn't get us to the progression in the chart. But never mind, let's just assume that in the first part of the Wikipedia article they transposed to C for simplicity. Here's another problem: the progression is described as falling major thirds or C, Ab, E, C (with an interpolated dominant). This progression is not unknown in Classical music, but usually in the form I, vi, IV, (V) I with first inversion chords interpolated. In diatonic harmony, this has a minor third and a major third. What I see in the progression given in the chart is, starting on B, up a minor 3rd, down a 5th, up a minor third, down a fifth, then a tritone, then a 4th. Also, the key seems to be G major, so the chords might be analyzed as III, V7 of I, I, V7 of bVI, bVI, ii, V7. I'm starting to see why in Jazz they don't analyze in terms of functional harmony!
So what John Coltrane has done is developed a harmonic structure with some interesting symmetries that tends to dilute the sense of tonic. I think the problems trying to analyze it start with the fact that in Jazz it seems to be the case that the idea of chord inversion is not significant. In other words, the idea of root movement is pretty weak. I say this because the bass and piano are free to play any element of the harmony as the lowest note. Similarly, so is the idea of cadence. The ends of sections are not marked with strong cadences--the closest we get seems to be a half-cadence. This suits the improvisational, open-ended character of Jazz.
But I still find it fundamentally unsatisfying because you can have open-ended music or you can have goal-directed music, but you can't really have both. So, forgive me, but what I hear when I listen to "Giant Steps" is some very cool music with a lot of character, but music that has no real direction. Intentionally has no direction, I believe, because the function of the non-functional, symmetrical harmonies is to remove that sense of real harmonic direction and movement that we find in Classical harmony. Fair enough. But when you take that away, what I hear is a jittery surface with no underpinnings...