"It is pretty extraordinary. But I don't feel special because of it. In fact I believe that it's more to do with Mirka, but the doctors might tell you otherwise."
One detail has yet to be resolved. Myla and Charlene are identical twins, fair-haired and dark-eyed.
So what about Leo and Lennart? "We don't know actually this time," Federer said. "For some reason, they couldn't tell if they were identical. So we are making a DNA test to find out."
A few hardcore fans have gone so far as to place bets on the boys winning Wimbledon, while one ingenious soul mocked up a scorecard from a mixed-doubles final in 2035: Leo and Charlene on one side, Myla and Lennart on the other.
"Yeah I saw that," Federer said. "It was funny. I got so many messages and congratulations. It's nice to see that people are happy for me, and especially for Mirka, because she did the hard work."
Federer has certainly been feted since his arrival in Rome. In the players' lounge yesterday, every single person offered congratulations. Rafael Nadal, rushing the other way, paused for a hand-clasp and a hearty clap on the shoulder.
The only one not playing ball was France's Jeremy Chardy, who knocked Federer out of the Rome Masters in the second round.
Federer, unusually, was sporting a pair of faint dark patches under his eyes. But then, judging by his own account, the sleep deprivation could have been far worse. Typically, for this family, Leo and Lennart are the perfect babies.
The hardest thing, Federer said, was deciding on what to call his lads. Asked when he had come up with their names, he said: "Maybe the day before. I must say boys' names was hard, girls' names was like this [snapping his fingers], even though we also didn't know until the girls were born what we were going to call them. We had to talk about it in the wake-up room.
"I feel like there's so many nice, beautiful girls' names, they're all cute and all that stuff, but with boys it's a totally different story."
Such issues are rarely discussed in locker-rooms. At 32, Federer has just become the first man since Ivan Lendl, in the early 1990s, to combine a full-time career with raising four children.
He has always said that he intends to keep playing while he enjoys the game. It would be a major surprise, given his love of the Olympics, if he did not compete in Rio in two years. But will the boys grow up with any memories of dad playing at the highest level? That might require him to go on to 36, or beyond.
"I'm aware it's going to be a lot of work," Federer said of his new life as the world's most famous father of four. "But this is not a time where anybody needs to feel sorry for me or get worried. It's super-exciting. With Myla and Charlene just being there and being with them, observing, just doing it all together, it's so cool ... the future is beautiful."
