Sorry, took a while to make a preliminary list of picks. I'm having some trouble fitting a handful of "Must Have" albums from the 70's here.
For our
first round pick, team
Twist/Shout picks
Raw Power by
Iggy & The Stooges (1973) in the
Punk category. (This is a bit of a stretch since "punk music" as a concept didn't exist in '73, but since it's one of the cornerstones of the whole genre, I don't see a problem.)

Sporting one of the greatest rock covers of all time, it's one of the more unlikely stories of rock'n'roll greatness.
The Stooges had broken up after two unsuccesful albums two years before, and after meeting producer Tony DeFries and David Bowie in New York (where much partying had ensued, ending in Bowie doing an intervention that propably stopped Pop from killing himself with heroin), Iggy Pop had been signed in Britain for Columbia records to make a solo album.
He had insisted on bringing along his old guitarist Alex Williamson, the original plan being that they'd find a rhythm section from overseas. However after some consideration they also brought in Ron and Scott Asheton for bass and drums, and so the Stooges was reborn in England. The band was pretty much left to it's own devices in the studio, resulting in something that Tony DeFries flat out refused to release, and Iggy was shuffled to LA while they figured out what to do.
In comes David Bowie, who was also signed by DeFries, to do a new mix of the album. Of course the budget was tight, so Bowie had all of one day to do it in an inexpensive LA studio. Initially this didn't seem to do much good, as the album was another commercial failure, and the band broke up again soon after.
However, the reputation of
Raw Power grew tremendously in subsequent years, and the album's volume and ferocity became benchmarks against which later albums were measured. In 2003, the album was ranked number 125 on
Rolling Stone magazine's list of
the 500 greatest albums of all time.
The first three songs from the beginning of the album, in order:
Search & Destroy
Gimme Danger
Your Pretty Face Is Going to Hell (originally titled "Hard to Beat")