Two decades ago, the new director of the Los Angeles County Museum of Art (Lacma) began a project to revitalize the space for the 21st century. On Wednesday, the institution unveiled the results of that $724m effort: the David Geffen Galleries, a hulking, curving concrete building that spans Wilshire Boulevard.

In a city of striking modern architecture, from the Getty Museum to the Disney Concert Hall, the opening is a landmark event. The project, whose unconventional shape has been likened to an amoeba, has inspired praise and polarization.

The 110,000 sq feet of galleries are housed in an imposing, elevated gray structure, ringed with enormous windows, that would be at home in an episode of The Jetsons. Speaking to the press on Wednesday, the museum’s director, Michael Govan, hailed the Swiss architect Peter Zumthor’s work as “of the future and past at the same time”, a building that is “accessible and sublime and new and different”.

The project has generated seemingly endless controversy since planning began in 2006; Zumthor’s proposal was revealed seven years later. The LA Times’ former architecture critic condemned its appearance – “limitless concrete” that’s “monotonous” – and its freewheeling approach to curation, and wondered about the practicalities of hanging art on concrete walls.

Some were concerned about the museum’s shrinking galleries – the Geffen Galleries have about 10,000 sq ft less space than the torn-down buildings that preceded it. Non-profit groups such as Save Lacma emerged amid concern over funding – $125m came from the county, the rest from private donors – and as several old buildings were destroyed. Zumthor and Govan butted heads over funding, and Zumthor suggested the process had been so difficult he would never work in the US again.

But the mood on Wednesday was jubilant, as Govan introduced the press to the nearly finished product, which opens to members on Sunday and to the public on 4 May. He likened a walk through the winding museum to wandering through a park or a forest, with works arranged thematically, not chronologically or according to any hierarchy, so that no artist is “in the back – everyone is in front”.

Zumthor, meanwhile, seemed far more enthusiastic about the United States than some of his previous remarks had suggested. Reflecting on what the country meant to him, the architect, born in 1943, described American heroism in the second world war, his relationship to the music of Miles Davis and Charlie Parker, and the shifting of the art world’s center of gravity from Paris to the US. After two or three weeks in Los Angeles, he said, “I had to admit: damn it, I love this city.” Being asked to design the museum as “an architect who made small buildings in Europe” was an honor, he said. Govan “trusted me. Thank you.”

The Geffen Galleries will showcase Lacma’s 155,000-piece permanent collection, with works in constant rotation. The galleries’ layout is indeed very different from that of a typical art museum, where works are arranged by era or movement and housed in separate rooms. The space instead feels defined by its openness, both internally and through windows to the outside. “There is no hierarchy here, no main entrance, no first floor,” said Naima Keith, senior vice-president of education, public programs and regional partnerships.

While some artworks sit in darker rooms, the majority are spread throughout a single, twisting chamber. An area focused on the American west contains images across three centuries, with a 19th-century landscape hanging near a mid-20th-century travel advertisement and 21st-century photography. Nearby, a young man balances a soccer ball on his head while moving slowly under it, and a trio of singers accompany him a cappella, all part of a “constructed situation” by the Berlin artist Tino Sehgal. A set of black-and-white images by the 20th-century Korean photographer Han Youngsoo is around the bend from divine Hindu sculpture dating back to 600AD. With few corners or flat walls, it’s easy to get lost among the art. A recent Los Angeles Times review – by a different critic – called the building “radically alive”, and other recent coverage has been positive.

“The idea was that it would be a reflection of the diversity of cultures of Los Angeles, that we could be a mirror of that,” Govan said in his opening remarks. “Nineteenth-century museums were a lot about categorization and knowledge, but we live in modern Los Angeles, where we’re all interconnected, where migration and interconnectedness are so essential to our daily life.”

The museum’s leadership emphasized the presence of the city itself in the galleries, where the outer walls are dominated by large windows treated to reduce UV light. (Many of the windows are also dimmed by light-filtering curtains, and the more light-sensitive artwork is kept away from them.) “You are not in a parenthesis of history outside of the present,” Diana Magaloni, the senior deputy director for conservation, curatorial and exhibitions, told the media. “You are there looking at the present and understanding the present as you look into the past.” Some of Lacma’s most prominent works are outdoors, including an enormous Jeff Koons topiary sculpture, Split-Rocker, that is covered in 50,000 living plants. Across the street, an Alexander Calder mobile perches in a fountain; it’s one of Lacma’s first commissions and echoed on tote bags across LA. The campus is adjacent to the Academy Museum, the Natural History Museum and the fragrant La Brea Tar Pits, pools that served as an inspiration for an early version of the Geffen Galleries’ “blob”-like architecture.

In the galleries, “you’re never fully cut off from Los Angeles”, Keith said. “Art and city are in conversation in ways that should have always been.” And if there were any doubts about the Geffen Galleries’ LA roots, the museum’s new cafe should put them to rest with an Erewhon collab. The presence of a trendy grocery store known for its $20 strawberries would seem to run counter to the museum’s democratizing mission; on the other hand, the museum will host free daily programs for families, no ticket required, Keith said.

Geffen Galleries “will feel like an invitation for students, families and our first-time visitors, and especially for the people who have always felt that museums weren’t quite made for them”, Keith said. “This building, this museum, is different. It’s designed to disorient you in the best possible way.”