The Beatles’ You Never Give Me Your Money and Pink Floyd’s Time (which has a reprise of Breathe at the end).
Have You Ever by** The Offspring**, it seems like two entirely different songs…
I was about to submit the Beatles’ You Never Give Me Your Money but typhoon beat me to it.
Steely Dan’s Aja is comprised literally of two songs that couldn’t stand alone by themselves but melded together make the perfect fit.
BNL’s Hello City has a bit of Housemartin’s Happy Hour in the end.
Goodbye Sky Harbor by Jimmy Eat World.
Elton John’s “Funeral For a Friend/Love Lies Bleeding”
Billy Joel’s “Angry Young Man”
Moody Blues “Question”
Someone else knows about my favourite band! Woooo!
Actually, “Suppose They Close the Door” is two songs. Chris writes in the liner notes to 4 Nights at the Palais Royale that it’s “made up of 2 recordings spliced together.” He had two incomplete songs when it came time to record Navy Blues, and the band just decided to make one song out of them.
“The James Bond Theme” is supposedly two tunes stitched together, as are Edgar Winter’s “Frankenstein” (thus the title) and Fleetwood Mac’s “The Chain.”
“Total Eclipse of the Heart” by Bonnie Tyler. My ex was like auntie em’s mom, she could never recognize the last part of the song from the first part. We used to play “name that tune” in the car, and I would get her all the time with this one, I’d name the title, and she’d say “no way!” only to recognize it later.
Alanis Morrisette has two songs that change rhythm mid-stream: Ironic and Uninvited.
“And When I Die” by Blood, Sweat, and Tears veers madly between two (or three, depending on how you count) rhythms.
At the risk of appearing to be a fan (which I am not, just happen to remember this song, I swear it!!)
Lionel Richie, “Say You, Say Me”
I’m a country gal myself, but back in the day (the high school day that is 
this was a popular song.
Can’t believe no one has yet mentioned Sweet’s “Love is Like Oxygen”, Tears for Fears’ “Head Over Heels”, Herbie Hancock’s “Rock It”, or Blondie’s “Rapture”.
Two from the genre of rap - Newcleus’ “Jam on It” and M.A.R.S.S.’ “Pump up the Volume”.
Doobie Brothers’ “Black Water” comes to mind.
They’re not related, but the doobie brothers!
“Heroin”: Lou Reed/Velvet Underground