We're all pretty familiar that traditional tonal music usually work within a tonic / dominant polarity. Holst works with this too; he does not completely abandon this idea. However, it is treated in different ways. I will try to illustrate this point using the movement Mars.
Ex. 1

Interestingly, Mars begins with a series of rhythmic pedal on G (Ex. 1). The melody moves in this way: G-D-Db. Subsequent melodic lines make use of this pattern. Looking at the key signature, I'm assuming this movement is centered around some sort of C tonality, but the establishment of this C tonality is quite ambiguous.

What lends this beginning special notice is the fact that the pedal G pattern (I'm quite unsure if I should use the term G / dominant pedal or just call it an ostinato pattern) keeps persisting for a large section of the beginning of Mars. Instead of using the dominant as a means of moving away from the tonic and subsequently as a preparation back to the tonic, as is the convention of traditional tonal works, the pedal G pattern at the beginning is used to resolve to the tonic, without even having the tonic established at first.
Ex. 2

Here at Ex. 2, the music is finally going into the tonic (aprox at 2.20min, refer to youtube clip posted by lovelove), as can be seen by the C bass (piano 1), which is further reinforced by that characteristic rhythmic pattern. A climax is built up here.

There're two things that happen here which could have a strong pull towards the tonic (C). First, the perrsistant dominat (G) pedal. Second, the leading note B, which occurs in context of a Bmaj triad rather than in a dominant Gmaj chord. There's an interesting play of tension here. On one hand, there is some relief on arriving on the tonic, at the same time new tension is created with the Dbmaj triad over a C bass, which makes it sound "not quite there yet".

I think one observation that I've made is that Holst has strong bass movements at certain parts which hint at some sort of key centre. However, what happens above the bass sometimes confuses people, especially when the melody moves in parallel triads (aprox 2.35min). In this way, Holst doesn't use diatonic harmonies (Ex. 3). There's also a play of mixture harmonies here (Amaj vs Amin between first two bars; Cmin vs Cmaj in last 2 couplets)
Ex. 3

Finally at Ex. 4, the music returns to how it started via a minor dominant 9th chord back to the rhythmic pedal on G (aprox 9.15min), but now with more power and aggression rather than as a means for building up the climax like before.

Ex. 4

And to round the movement off, Holst decides to end on C, through the repeated C bass (Ex. 5).
Ex. 5

So if you take a look at the big picture, this is a rough outline of the key structure from beginning til end, in the context of C tonality (not particularly C major / minor). It's quite unusual considering how Holst begins on the dominant and ends on the tonic.

G (Ex. 1) - C (Ex. 2) - X (exploring other keys) - G (Ex. 4) - C (Ex. 5)