Harry Chapin got famous wistfully narrating his semi-autobiographical story-song with a bit of a world-weary growl in TAXI, which I like to think involved audiences shrugging off the brief weird falsetto interlude from Chapin’s bassist about flying and crying and dying.
“Down With the Sickness” by Disturbed has a big chunk, a rant by lead singer David Draiman against his “mommy”, that doesn’t fit with the rest of the song.
“The Battle Of Epping Forest” by Genesis interrupts the “battle” portion of the song to tell the story of “the Reverend” before returning to the “battle”.
Jay and The Americans’ version of “This Magic Moment” has a part in the middle with a brass fanfare for no reason.
And in “God Only Knows” an ethereal ballad is interupted by an instrumental break that starts like a rumba lesson.
“Blow” by Kesha (in my opinion, the best song of the year) has a very strange bridge. The music changes quite drastically, and she starts chanting, “go insane, go insane, throw some glitter, make it rain on 'em. Let me see them hands, let me let me see them hands.” Also, the way she says “hands” is indistinguishable from “Hanes” (as in Hanes underwear). The video also has a bizarre conversation about Muenster cheese between Kesha and James van der Beeck (who she calls “James van der Douche.”)
ELO’s Evil Woman has a searing orchestral strings moment or two about two-thirds of the way through. Completely at odds with the rest of the song, but I guess it’s been in my memory so long that it would seem odd (wrong?) to have it left out.
One can almost see the villainess, in a black-and-white movie made before the addition of sound, caught in the sudden floodlight of discovery, her heinous deeds exposed, her face panicked as she cowers before turning to flee.
Uhm…so I’ve heard. 
Several Guns N Roses songs seem to change into something different near the end. For instance, November Rain.
If I remember correctly, it’s inaudible on the old 45rpm version, audible on all others.
Yardbirds “for your love.” Clapton quit part way through the recording, so the middle is all filler.
The “James Bond” intro to the Beatles’ “Help”
The original release by Soft Cell segued into “Where Did Our Love Go”.
“You Don’t Have To Say You Love Me” by Dusty Springfield opens with a bombastic orchestra and chorus before settling into the song proper, which is considerably softer.
At least part of that string passage is a backwards track – it’s a measure from Nightrider, the next track on the album (“Face the Music”), played backwards.
That’s gotta be it. Having noticed the similarity, I’d have heard if they were stepping on what I thought of as MY punchline!
Here, incidentally, is the ad:
Wild Thing, the Troggs. What fits better into a super primitive rocker than an ocarina solo? Hell does a ocarina solo fit in any song?
Whole Lotta Love, Led Zeppelin. That weird midsection of theremin and studio trickery has nothing to do with the the primal blues of the rest of the song. Yet somehow when it’s edited out the song loses a lot of power.
Although this doesn’t perfectly fit, Jethro Tull stuck a goofy little kids’ story “The Hare Who Lost His Spectacles” right in the middle of its epic album length song “A Passion Play”.
There are songs that entirely consist of anomalous parts, the one that springs to mind is The Most Unwanted Song (youtube this 22-minute long monstrosty [I mean that in a good way!] if you don’t know it already.)
While not a weepy-romantic song, this reminds me of Memorial Day by Carly Simon. At the end of each chorus of this medium-slow song, the music veers from epic light rock (if such a thing can be imagined) into orchestral fast paced disco music, only to die back down again by the next verse.
This was never very popular, but I’ve always liked Rick wakeman. On his Myths and Legends of King Arthur and the Knights of the Round Table album, his composition Merlin mixes some very different music styles, starting with a chanting chorus, going to soft keyboard, then rock. But it REALLY gets weird just short of 5 minutes into it, when it goes into full Looney Tunes mode for about 40 seconds, then goes back to what it had been doing, lulling you into a false sense of security so it can surprise you with Looney Tunes at the end again.
Everything ever recorded by Rush.
But the best example has to be Roll The Bones, which has a passage in the middle which is …
a RAP!
How about the kazoo ensemble in “Johnny Get Angry”?