Quote: ...Which made me look up quickly what Stravinsky wrote that was atonal, and came up with "Firebird" (my internet source was wrong, because it is not atonal. But it does seem complex harmonically).
...So I went to my Stravinsky box set, "The Complete Columbia Album Collection:"
Wait, the internet was wrong about something?!?!

.
I would characterize most of what Stravinsky wrote as expressionistic and chromatic, so at least atonalish, but definitely not serial 12-tone. Meaning it was mostly based on triadic harmony, just not in clearly defined keys. Same as e.g. Bartok or Ravel. But at the end of his life he did get around to writing (more or less) 12-tone, atonal music. However being Stravinsky he put his own spin on it and it sounds very different from e.g. Schoenberg's 12-tone compositions. The one I know and like is his ballet Agon.

On CD.
A lot of people swear by Stravinsky's recordings of his own music, but I agree with Leonard Bernstein: Stravinsky was a great composer and a bad conductor. For Agon I like Robert Craft's recording on Naxos.