Four South Africans, including the twins, Brandon-Lee and Tony-Lee Thulsie, faced charges in court ranging from conspiracy to firearms offences, the spokesman for the elite police unit Hawks, Brigadier Hangwani Mulaudzi, said.
The four, arrested in Johannesburg on Sunday, will be detained in custody until July 19, when their case will be heard, Mulaudzi said.
Quoting the charge sheet, the News24 news organization said the twins had been attempting to fly to Syria. Security officials say there are no known militant groups operating in South Africa, but Britain and the United States warned in June of a high threat of attacks against foreigners in the country's shopping malls.
Mulaudzi named the other two siblings as Fatima and Ibrahim Mohammed Patel.
"The indictment does talk to issues of terror plots that they were planning against the US Embassy as well as Jewish Buildings in the country," he said, referring to the twins.
"The twins have been charged with conspiracy," Mulaudzi added. "The Patel siblings have been charged with the violation of the Firearms Control Act for now."
The twins' preliminary charge sheet states that their conspiracy occurred between October 2015 and July 8 this year, local newspaper the Times said on its online service.
In Washington, State Department spokesman John Kirby said at a daily news briefing that the United States applauded Hawks for making the arrests and had "full confidence in the South African judicial system to handle this case according to internationally accepted best practices".