This song is mixolydian. If you're talking about the chords, you can't really use the normal numbers to identify them because...well, the five chord is minor, for example.
And that five chord is used in this song - very cleverly, I might add! It's really used as a color variant of the flat seven chord. You've got a verse where E is the tonic and it alternates back and forth between an E major chord and B minor. In the section that follows it (I don't know what you'd call it - a pre-chorus perhaps, but there really is no chorus), it's going back and forth between E major and D major instead. I definitely feel that these two different means of moving away from and back to the tonic are variants of each other and that the B minor is this sort of colorful harmonic variant of just going back and forth between a tonic and a flat seven.
Now, in that same second segment, after going back and forth between E major and D major a couple of times, they go to an A major chord. If we're talking about these as diatonic chords, that's a tonic I chord. Clearly, though, we've established E as a modal tonic, so it's more like a IV chord.
What is the function of a IV chord, though, in the mixolydian mode? There's no dominant chord, so it can't be a dominant prep.
The Bangles play with it. The first time through, they go from A up to a B major dominant chord as a bit of a surprise (briefly changing the modality). The second time through, at 1:23 in the video, they don't. Again, color variants.