It looks like the chorus in Lenny is an example of parallel harmony or parallelism. The first chord (the A6) is repeated with different root notes, but otherwise with the exact same intervals and even the same voicing.
Arguably the modern godfather of parallelism is Claude Debussy. When asked by his teacher about the "theory" of parallelism, Debussy replied, "There is no theory. You have merely to listen. Pleasure is the law."
So parallel chords don't really admit themselves to "normal" analysis. One way to get a grip on them is to look at the melodic motion of the root notes and see them as almost single-note runs that just have a very complex timbre (sort of like a stacked 5ths monosynth patch, or a set of organ stops that emphasize the partials over the fundamental). The PDF linked above describes it as "a melodic line amplified by a chordal layer". And later on in the PDF (emphasis mine):
Such instances of parallel chords are certainly coloristic, but it is important to recognize that they nevertheless are essentially melodic events, whose specific harmonic function is suspended; moreover, they are guided by the chromatic scale, almost never the diatonic.
In the case of Lenny, the G and D naturals firmly take us out of E major. My interpretation is it in a minor mode of A, perhaps Phrygian or Aeolian, more likely the former in light of two B flats and only one B natural. Then we come out of the chorus with IV-I plagal cadence back to E major.