KEXP uploaded videos from the Zombies' set at SXSW. I'm not sure whether it was just because I hadn't listened to "I Want You Back Again" for a long time or because I could watch Rod Argent, but it became obvious to me that the chord progression was based on fifths. So I figured it out. It's mostly just C minor and F major.
I think that makes "I Want You Back Again" one of the few songs where I learned the piano part before anything else. This is a bit rough, and I play the wrong notes in a few spots, but I just learned it to-day. The blank part is where the solo should be.