I had always encountered the prayer in traditional renderings with traditional patterns of speaking it. According to those patterns the prayer opens with two thoughts
1. Our Father in heaven, hallowed be your name
2. Your kingdom come, your will be done, on earth as it is in heaven
I always took "on earth as it is in heaven" to qualify both "your kingdom come" and "your will be done." However, the revelation for dim-old-me was that it also qualifies "hallowed be your name."
The prayer is three requests that all run in parallel
Your name be hallowed [on earth as it is in heaven]
Your Kingdom come [on earth as it is in heaven]
Your will be done on earth as it is in heaven
I had missed this insight because:
(a) the traditional English translations obscure the parallelism by having a different word order in the first request from the second and third. (In Greek all three are parallel.)
(b) the rhythm of the traditional English performance of the Lord's prayer makes a clear distinction between the first request and the next two. (Just say it to yourself and you'll see what I mean.)
I thought that this was quite interesting.
It is also always worth reminding ourselves that "hallowed be your name" is a request that God cause his name to be hallowed and not, as I saw on one recent attempt to put the prayer in contemporary language, the equivalent of "Praise your name!"