But neither side has shown much taste for reconciliation. Islamists staunchly reject the new leadership and insist the only possible solution to the crisis is to put Morsi back in office. Meanwhile, the interim leadership is pushing ahead with a fast-track transition plan to return to a democratically elected government by early next year.
And the military-backed authorities appear confident of public support for a tougher hand after millions turned out for nationwide rallies Friday called by army chief Gen. Abdel-Fattah el-Sissi as a mandate against "terrorism and violence."
Interior Minister Mohammed Ibrahim, who is in charge of police, took an uncompromising stance in a press conference after the violence. He accused the pro-Morsi side of provoking bloodshed to win sympathy.
"We didn't go to them, they came to us — so they could use what happened for political gain," he said. Ironically, Ibrahim is originally a Morsi appointee, and his then-boss praised him for a tough hand after police killed dozens of anti-Morsi protesters in the city of Port Said earlier this year.
"The Ministry of Interior never has and never will fire on any Egyptian," he added, saying police only fired tear gas in Saturday's violence.
He suggested authorities could take the more explosive step of moving against the two main pro-Morsi protest camps in Cairo: weeks-old sit-ins, on outside the Rabaah al-Adawiya in eastern Cairo and another in Nahda Square in Cairo's sister city of Giza.
He depicted the encampments as a danger to the public, pointing to a string of 10 bodies police have said were found nearby in recent days. Some had been tortured to death, police have said, apparently by members of the sit-ins who believed they were spies.
"Soon we will deal with both sit-ins," Ibrahim said.