As the self-driving car navigated the narrow, winding streets leading to San Francisco’s Coit Tower, Jesse Levinson narrated from the back seat like a proud parent.

“Isn’t that cool, how we slowed down for that person and then nudged around him?” said the chief technology officer of self-driving startup Zoox. “See how steep this street is? We cover all the most challenging terrain.”

Of the dozens of Bay Area companies racing to make robot cars, Foster City’s Zoox may be the most secretive.

This week it revealed plans to provide free autonomous rides (with a safety driver, as required by California law, and another engineer in the navigator’s seat) between the Fairmont San Francisco hotel and Moscone Center for select attendees of the Global Climate Action Summit. The point: show off its cars’ agility in complex urban environments and underscore the social and environmental benefits of its forthcoming all-electric robot rides — although this week’s rides will be in retrofitted hybrid Toyota Highlanders.

Founded by two relative unknowns, Zoox isn’t backed by a major auto company like General Motors’ Cruise, or by a major tech company like Waymo, Baidu, Apple, Uber or Lyft.

But it has raised $800 million, has a private-market valuation of $3.2 billion, has attracted almost 600 employees and says it’s on track to offer a robot taxi service in 2020 in San Francisco.

Zoox sprang up in 2014 as the brainchild of Australian artist-designer Tim Kentley-Klay, who had no tech background but was excited about self-driving cars’ potential, and Levinson, then a postdoc in computer science at Stanford.

The brash salesman and brainy engineer combo sounds like another well-known Silicon Valley pair, but Levinson deflects comparison to two Apple co-founders, Steve Jobs and Steve Wozniak: “We haven’t created a trillion-dollar company yet.”

Zoox has echoed Apple in at least one way, though. Just as Jobs was forced out of Apple in 1985, last month Zoox abruptly axed Kentley-Klay as CEO.

Levinson, who has added president to his chief technology officer role, is temporarily running the company alongside board member Carl Bass, the former CEO of Autodesk, who is serving as executive chairman. Zoox is seeking a new CEO.

“We’re tremendously grateful for Tim’s vision and insight, and everything we accomplished over the past four years,” Levinson said, adding that none of Zoox’s employees left.

“We are continuing on the same mission as before,” Bass said. “There’s been some fake news about the company changing direction or being sold; none of that’s true.”

Kentley-Klay did not respond to an email requesting comment. In a since-deleted August tweet, he wrote that the ouster showed “Silicon Valley up to its worst tricks.” His Twitter feed remains filled with unattributed quotes, seemingly from Zoox employees, expressing shock at his departure and offering support.

Zoox’s outsize ambitions — it plans to build its own radically reimagined cars from the wheels up, rather than retrofit existing ones — have fueled rumors of a sale to a deep-pocketed company such as Apple, or one with manufacturing expertise, such as an automaker.

“As Tesla has shown, manufacturing cars is difficult,” said Jim McPherson, a Benicia attorney and new-mobility analyst. “Zoox may have grand plans, but to manufacture at scale without defects, it’s probably better off partnering with a company that’s already building cars.”

The industry views Zoox as a dark horse, McPherson said, compared with companies like Waymo that are on twin tracks of development and public relations as a way of building confidence.

Zoox’s demo rides this week in the retrofitted Highlanders may draw some attention, but it won’t take the wraps off its car design. A prototype at its San Francisco office is inside a giant locked crate.

“Don’t feel bad; we haven’t let any other press see it either,” Levinson said. Only investors get a peek. “We figured people might not want to give us hundreds of millions of dollars without seeing what our vehicle looks like.”

“We’ve created a new type of vehicle unlike anything anyone has experienced before,” he said.

The front and back will be symmetrical, and the car will be able to drive in either direction with ease. Each wheel will have its own motor, with sensors and cameras integrated. The four passengers will experience “social seating” in which they face one another as if around a dining table. A battery will power the all-electric vehicle all day.

Custom lights and sounds will communicate with pedestrians and other cars, such as notifying someone who is in the way. “It will be inclusive and multicultural so as not to shout English words at them,” Levinson said.

Kentley-Klay and Levinson were introduced by Anthony Levandowski, former star Google engineer who became infamous as the center of a bitter trade-secret lawsuit with Uber involving self-driving technology. (Levandowski has never been involved in Zoox, Levinson said.)

“Tim and I started off with essentially nothing other than a vision,” Levinson said. That quickly changed.

Levinson had gone to Stanford to work with Sebastian Thrun, considered the father of self-driving cars.

Stanford granted Zoox an exclusive license to technology Levinson and his team developed, including some that was part of the 2007 DARPA urban challenge, in which self-driving vehicles competed to safely navigate city streets.

The company, founded in 2014, raised $40 million within a year; within two years, another $250 million.

“Other than in biotech or pharma, it’s unusual to start a company where the first product will be six years out,” Levinson said. “Because the scope of what we are trying to accomplish is so exciting, we were able to find top-tier investors who saw the size of the opportunity and were willing to place money on something not without a degree of risk.”

Levinson, the son of legendary Silicon Valley figure Arthur Levinson, the chairman of Apple and former head of Genentech, said his father’s connections played no role.

“The only introduction I ever had through him was to one investor who didn’t even want to meet with us in 2014,” he said, declining to name that investor.

Carolyn Said is a San Francisco Chronicle staff writer. Email: csaid@sfchronicle.com Twitter: @csaid