INXS - Need You Tonight and Mediate
Despite the two titles, that is really one song. They do not really work separately (though together they work brilliantly).
Another dual-titled song that can’t really be separated into two without huge loss is Won’t You Try/Saturday Afternoon from Jefferson Airplane’s album After Bathing at Baxter’s (although it may not count as “commonly played”). Although there is a rhythmic change, lyrical and melodic motifs from the first “song” continue to be woven into the fabric of the second, so that it is really impossible to say where one ends and the other begins.
Helter Skelter and Long, Long, Long on The Beatles’ White Album are, by contrast, quite distinct songs, but the segue between them, the sudden but seamless descent from the raucous noise of the first to the calm softness of the second, is terrific. If they aren’t commonly played together they should be.
Another great segue is that between China Cat Sunflower and I Know You Rider on the Grateful Dead’s Europe '72. This is more of an instrumental bridge than a direct segue, but I think it is perhaps the best few seconds on the whole triple album.
JHC I got two out of the three wrong.
Aquarius / Let The Sunshine In (both from Hair) - The 5rh Dimension
Also “Threshold” and “Jet Airliner” by Steve Miller Band. It’s always exciting finding out which version of the latter the DJ plays: is it the one with the line “and I don’t want to get caught up in any of that funky kicks going down in the city?” Or is it the one with “…funky shit going down in the city?”
Are you sure you don’t mean Sgt. Pepper’s Lonely Hearts Club Band (Reprise) / A Day In the Life ? 
ETA: dammit, I hate it when I forget to refresh before posting!
Ziggy Stardust/Suffragette City – David Bowie
ZZ Top: Waiting for the Bus/Jesus Just Left Chicago
And of course, from the same album, “Stone Love”/“Moonage Daydream”.
Falling In And Out Of Love With You / Amie - Pure Prairie League
Rambling On My Mind / Have You Ever Loved A Woman? - Eric Clapton
This wasn’t strictly speaking two songs played back to back. It’s a single mix comprising the two songs.
Your move/ I’ve seen all good people. Yes
This Beat Goes On / Switchin’ To Glide - The Kings
Devil With a Blue Dress On / Good Golly Miss Molly - Mitch Ryder and the Detroit Wheels
Brain Stew/Jaded - Green Day. Usually.
Devo had a few of these…
Gut Feeling/Slap Your Mammy
Smart Patrol/Mr. DNA
Timing X/Clockout
I love this answer.
Another Nail In My Heart/ Pulling Mussels (From The Shell) by Squeeze.
Simon and Garfunkel: Save the Life of my Child/America
Jethro Tull: Thick as a Brick, side 1/Thick as a Brick, side 2
(or is that cheating?)
Bob Seger: Travelin’ Man / Beautiful Loser
J. Geils Band: Whammer Jammer / Hard Drivin’ Man
mmm
The Star Spangled Banner/Purple Haze/Instrumental Jam.