Imagine you’re in a tavern and this charismatic guy at a nearby table who you wish could be your new best friend—call him best friend #1—is talking loud enough for everyone to hear about the amusing things he’s just been doing, and then as his narrative gets to its peak another appealing guy at an adjacent table, your best friend #2, who in reality is best friends with best friend #1, chimes in to finish #1’s lines with perfect timing—a magical eavesdropping experience.
This is the closest similitude I can think of for listening to “I Can’t Sleep.” Lee Mavers sings with himself on those marvelously complementary low and high notes in the last section of the verse, which makes the back-and-forth of it seem not false but all the more charming, like he’s successfully turned into two people (and who has never wished for this?) to have himself for a buddy.
“I Can’t Sleep” first verse
I was looking through my window
I was looking through my eyes
And there’s a big black* car coming
Get off the street out*side—come in
Go into a pipedream
No one listens anyway
When I say I’m so (so) sad about
Not so (so) glad about
I’m so (so) mad
I can’t sleep tonight
*Notice the wonderful deep tom strike here
Using the tonic as home base, he’s singing up to the fifth, the sixth, and seventh, those intervals tracing in the opposite direction the I, IV, and flat-VII chords that repeat throughout the song. The intervals are big enough to get him from his gravelly chest voice up to his clearer-sounding head voice. (I’m not sure if this chest/head distinction applies to Mavers’ singing, but it does to mine; anyway, he gets that sort of effect.)
This is one of the very great, very simple ideas. Like everything else in The La’s songs, it sounds archetypal. Where else have we heard it?
 He overdubs this on the recording, but you can hear him singing it much the same way live.)