By all accounts, he was a doting dad to his year-old twin son and daughter, setting up a bouncy castle in the yard for their recent first birthday party and dressing them in their latest cute outfits while his wife made breakfast in their Rockland County split-level.

Then, on Saturday, Juan Rodriguez, 39, was hauled handcuffed before a judge, charged with the babies’ hot-car deaths.

He forgot to drop them off at their day care Friday morning, Rodriguez has told cops — and didn’t realize they were still in the back seat when he parked at the Bronx hospital where he’s a social worker, court papers reveal.

Eight hours later, the twins, Luna and Phoenix, registered an internal temperature of 108 degrees when coroners examined their bodies in their car seats.

“I assumed I dropped them off at day care before I went to work,” Rodriguez told cops at the scene, according to the criminal complaint against him.

“I blanked out!” he cried.

“My babies are dead! I killed my babies!”

Rodriguez was still sobbing at his arraignment Saturday night before Bronx Criminal Court Justice Patsy Goldborne.

A disabled Iraq war veteran, he still wore the same turquoise polo shirt he’d worn Friday as he cared for homeless and ailing vets at his job at a VA hospital in Kingsbridge — oblivious that his twins were in the back seat of his Honda Accord in the parking lot.

“He carried on with his day,” Assistant District Attorney Jaime Breslin told the judge.

“He forgot his children in the seats.”

“This is a tragedy of horrific proportions,” his lawyer, Joey Jackson, told the judge, who set bail at $100,000.

Two hours later, Rodriguez made bail and was released to his cheering, weeping family members.

Rodriguez exchanged a tight embrace with his crying wife, Marissa — the twins’ mother — when he was released.

The mother had come to court wearing all black, and with the couple’s surviving child, a 4-year-old boy, in tow.

The dad had dropped the boy off at a different day care before forgetting the twin babies, officials said.

The mom held the boy on her lap as she waited in a second-floor holding area for the arraignment to begin. With her were some two dozen family members and friends.

At one point, the mom broke into loud, hysterical sobs as she embraced a female relative.

At another point, Rodriguez’s mother, Cathelina Valerio, cried out, “Mis nietos! Mis nietos!” (“My grandchildren!”) as she hugged a male family member.

“Do you know what I’m scared of?” she told The Post. “When he gets out, he can’t handle it. He’s going to need help.”

It was a horrific double death — a baby brother and sister, side by side and helpless, as the temperature in the parked vehicle spiked.

It was also a tragedy made inexplicable by Rodriguez’s reputation as a caring father.

How could a dad whose social media is crammed with photos of him snuggling with Luna and Phoenix — and whom neighbors universally describe as loving and attentive — have forgotten his twin treasures, all day, in the back seat of his car?

“This was just a horrible mistake,” neighbor Tony Caterino, 45, said of Rodriguez. “That one time you make a mistake, and you have to live with it for the rest of your life.”

On Friday morning, Rodriguez had driven to work, as usual, from his home in New City, a middle-class, heavily wooded suburb an hour’s drive north of the city.

Video shows it was 8:22 a.m. when he parked in the hospital parking lot, police sources said.

The video shows him returning at one minute before 4 p.m.

Rodriguez started the car and drove north toward home — only to pull over less than 10 minutes later, while still in the Bronx, on Kingsbridge Terrace.

“I left them in the car!” he began screaming.

Witnesses called 911; arriving medics could not revive the babies and they were pronounced dead at the scene.

Family friend Temple Barros, 41, who lives with the family, told The Post that Rodriguez routinely took the twins to a day care in the mornings.

The twins would stay there throughout the day, Barros said, as Rodriguez was at his hospital job and Marissa worked as a tourism and travel sales manager at the Empire City Casino in Yonkers.

And while Barros didn’t see Rodriguez and the twins leave the house Friday, it would have been the dad, not the mother, who put them in the car, he said.

“An amazing guy,” Barros told The Post of Rodriguez.

“He’s always been there for his kids. Always. This is just a horrible situation,” he said. “The family isn’t doing so well,” he added.

Rodriguez was “always doing things with the kids,” said Caterino, the neighbor.

“Always in the yard playing. They had a big campout last weekend, with tents in the back yard.

“He would always play catch or basketball with his older kids,” Caterino said of Rodriguez’s two older sons from a prior marriage.

“I just can’t wrap my head around it,” said another neighbor, Paul Barlett, 39. “No one here can believe it.”

Additional reporting by Stephanie Pagones, Kenneth Garger, Larry Celona and Laura Italiano