There appears to be this phenomenon where a band, a band that tends to prefer rocking, has a mega hit with a ballad that seems completely a-typical to their usual rocking selves.
Off the top of my head: "No Rain" for Blind Melon, "More Than Words" for Extreme (not on this list, but a prime example), maybe even "Losing My Religion" for REM.
What's weird about "Every Breath You Take" is that, while it appears to be a false-advertising deviation from the usual rocking the Police liked to do, it is actually a perfect extension of everything that band liked to do all the time.
It starts with drums. To me, the high delivered straight to my brain every time I hear the Police is the drum work of Stewart Copeland. First of all, his name sounds like he was born holding drum sticks that is such a drummer-sounding name. Copeland stands in a long and illustrious line of drummers I like to call the "agile drummers." There is sharp attack and frenetic movement in everything they do; they own all regions of their kits; and even if they are just holding on a hi-hat/cymbal, snare, kick pattern, there is constant rhythmic attitude and staccato greatness to enjoy. Preceding him in this line was Mitch Mitchell of the Jimi Hendrix Experience, following him was Matt Cameron of Soundgarden (also the permanent Pearl Jam fill-in).
In this song, Copeland has almost nothing to do at first - yet he still does so much in its service. His cinderblock-solid snare drum crack sets off the song, then (without any hi-hat/cymbal accompaniment) holds the ideal backbeat, echoing into the available silence. Sharpness providing air, air becoming mystery.
Meanwhile, the legendary guitar pattern of Andy Summers is just what he always brings to every song. He had an interesting role in this band, basically a second fiddle to Sting's lead bass. He could afford to form these excellent accompanying patterns that travel all over the fretboard without ever treading into the territory of taking the lead. He is doing nothing different on this song, except the dynamics are low enough that he almost accidentally shines through more than normal. Suddenly this his signature guitar pattern simply because Copeland and Sting are not flying all over the place in front of him.
The bass, in the verse, is still kind of the heartbeat, even just hitting root notes and doubling the guitar rhythm. The fact that Sting is backing up the guitar lends maybe some subtle credibility to the guitar part. He only slightly varies his picking rhythm in small moments to counterpoint the guitar, because Sting can't let himself be totally invisible.
Then there is the Sting vocal. Butter. Questionable sanity butter, but butter. He kind of set the standard for the pretty British tenor, didn't he?
What is the chorus? There is some formal ambiguity here. The chorus I think begins with the line "Oh can't you see," but it is a very downbeat chorus. It functions more as an prologue to the real event, this magnificent post-chorus bridge. Songs that fry formal expectations but don't sizzle too loud about it have a real chance to be their own songs.
That post-chorus bridge is maybe the most mature sounding sequence in the Police discography. The Police always found a way to employ piano in great supporting roles, and here it is just what needs to happen, these tones ringing out after each dramatic guitar blast. The vocal: Butter.
Notice here how readily the drums take on a greater role but keep their consistent sharpness. The cymbals of Stewart Copeland's are made of some kind of elfin metal, so sharp, so bright, occupying a targeted frequency with the magical efficiency of mithril.
This is, in short, the Police. This is what they do. It is not a performance outlier, as maybe the 538 would call it if the Police were a sports team. The technical consistency underlying much of their music is pretty impressive, if nothing else. In this case, it just happened that all the usual ingredients shook out into a strange, unique, darkly obsessive ballad, and the result took over the world's ears.