Here is a poor quality substitute for the fantastic video that seemed to have just been removed above. It depicts early rehearsals/arrangement for the Stones sympathy for the devil. In youtube comment section you can jump to various sections, such as an unrelated live concert of the song near the end. They mercifully omit the 35 takes needed for the actual recording. For some reason I am more interested in historical human drama than the song, and will yak a bit about it below the video.
The docu-filming was by famous Jean-Luc Godard for a dud movie. It's amusing to see some of the stones and their girlfriends self consciously overdressed for the occasion. The session piano-god Nicky Hopkins seems dressed and haircutted for church instead of his usual homeless look (partly from digestive disease that would kill him). The blond founder of the stones seems disengaged and patronized with possibly guitar mike switched off, and on his trajectory towards being fired and subsequent death. Too bad because in early songs he contributed really catchy embellishments.
The girlfriends who kiss and cuddle while helping with the chorus aren't just bystanders. The caped one inspired the lyrics due to a Russian novel she gifted to Mick. The other girl has famous involvement with the founder - was she the one he chronically beat up and had to be kind of re-assigned to another stone? Anyway the caped one suggested the signature woo-woo chorus, and that and conga drums finally elevated a kind of limp lamentation song to brash pop immortality.