They had their differences. He was into restoration; she preferred new. She wanted to keep the original façade and floors, he wanted to redo them. He won that battle.

But Ms. Greenspon has won others. The two are business partners, not romantic partners, and Algus Greenspon, their West Village art gallery, has the smooth cement floors he wanted but also shows some of the young artists she prefers.

Even by art world standards, it is a peculiar arrangement. What has made it work?

The two first spoke around 2000, when Ms. Greenspon was at NYU and working for the PaceWildenstein gallery. She contacted Mr. Algus because she was interested in buying a painting by the obscure Pop artist Nicholas Krushenick, whom he had shown. “I remember calling, and someone answered,” she said. “They were like, ‘Mitchell doesn’t come in until after 4.’ I thought, ‘O.K.. This guy must be really important.’”

A lot of people—high-profile artists, curators and writers—really do hold Mr. Algus in high regard. But the real reason he wasn’t at the gallery was that he was at Long Island City High School.

Mr. Krushenick was not the only obscure artist Mr. Algus showed. There were also Larry Zox, Harold Stevenson, Rosalyn Drexler and dozens more. From 1992 until he joined with Ms. Greenspon, he specialized almost exclusively in the work of unknown or forgotten artists from the 1960s and ’70s. He was heralded for reviving careers, sometimes to his own detriment.

“It’s the ones like Lee Lozano who get upsetting because you realize how powerless you are,” he said on a recent late afternoon over iced coffees at the Eli’s Essentials café next to the Whitney Museum. He’d just finished teaching. One of the leading conceptual artists in New York in the 1960’s, Ms. Lozano quit the art world and was virtually unknown when he and other dealers presented her work in 1998, shortly before her death. Museum retrospectives followed those shows, and international powerhouse Hauser & Wirth now works with her estate.

“We were trying to sell a painting for $18,000 that would go for over a million now,” he said. “Money is sexy. A painting that is a million dollars is a lot more sexy than a painting that is $18,000. It’s very frustrating. You realize that is the way the art world is.”

An article in this month’s Forbes discusses today’s high-flying art market and talks about dealers being celebrities. Most aren’t.

“The art market,” Mr. Algus said at Eli’s, “has never meant anything to me.” Dealers often say this, but in Mr. Algus’s case it appears to be true. Over two decades he has run four relatively quiet galleries in four different neighborhoods, while by his own admission selling very little.

Selling was not really the point. It was about getting under people’s skin, aesthetically speaking. “I really dislike consensus,” he said. He propped his glasses on his head and continued emphatically, “When everybody agrees on something I just feel like it’s all the nasty kids I hated in high school, and so, if things can undermine consensus, that’s sort of what I’m interested in.”

That would be South Side Senior High School, in Rockville Centre on Long Island. “He had these swept bangs,” recalled artist Deborah Kass, a fellow student. “He was fucking adorable.” Mr. Algus is pretty much bald today, but still radiates boyish enthusiasm. At Eli’s, The Observer admitted unfamiliarity with the painter Louis Eilshemius. “You don’t know Louis Eilshemius?” He gaped in horror, then pulled up JPEGS on his laptop.

Art wasn’t originally in the cards for Mr. Algus. Though his father bought art magazines and took him to galleries, he studied geology at Harpur College, now SUNY Binghampton, and got his master’s in the subject at McGill University in Montreal. “I really wanted to go up to the Canadian Arctic. That whole British ‘to the ends of the Earth’ tradition was really alive there.”

He went to the University of Colorado-Boulder to get his doctorate and found himself hanging out in the art history department. But, he said, “It was very pretentious. It was really intellectually rigorous and very dry.” He went back to McGill and got his Ph.D. in geology.

In 1986, he moved back to New York to teach grad school, and he and a friend opened a gallery in Williamsburg. He was also making art—small assemblages he exhibited at East Village galleries.

Williamsburg was uncharted territory. “We would have an opening and there would be 500 people out in the street,” Mr. Algus said. They showed their friends’ work. “We would have a bonfire, and then no one would come again. It was just a party, and then a month of nothing.”

One day Mr. Algus’s girlfriend, Dyanne (who is now his wife), had her wallet stolen from her car. A man who’d done odd jobs for the gallery learned the thieves wanted $40 for it. “Then I get a phone call that he got into a fight with somebody and stabbed him and he needed a plane ticket to Puerto Rico and he was living in the basement hiding out from the police. So, the gallery was over.”

His next gallery, which he opened about nine months later, in November 1992, was on Thompson Street in Soho. There he began dusting off artists he’d read about in his father’s old art magazines. He stopped teaching graduate school and moved to a high school so that he could head to the gallery in the late afternoon.

He opened with The New Adam, a 39-foot-long nude portrait of Sal Mineo that painter Harold Stevenson—a well-regarded but then-unknown artist—made in 1962. The painting, which was subsequently donated to the Guggenheim, took up his whole gallery.

“He was like a fairy godmother for us,” Betty Tompkins said in her Soho loft recently, surrounded by a few of her “Fuck Paintings”—huge, virtuosic black-and-white airbrushed canvases of heterosexual penetration—that Mr. Algus showed, ending a decades-long drought in her career, which subsequently took off. Her first “Fuck Painting,” from 1969, is now owned by Paris’s Centre Pompidou.

Unlike at many galleries, women were integral to Mr. Algus’s program. There were Ms. Lozano and Ms. Tompkins, but also Joan Semmel, who paints fleshy full-body portraits, often in lurid colors, and Judith Bernstein, who’s known for brawny drawings of flying penises that sometimes cover entire walls.

All of those artists were mid-career or well past it when Mr. Algus entered the picture, and all now work with other galleries. When his artists moved on, they generally did so with Mr. Algus’s blessing. “It’s good, because I don’t want to represent people in that way,” he said. “It takes the pressure off me.”

With a small space, low overheard and few employees, he could be discerning in how he operated. “I was actually there at the gallery one time, and he’s on the phone with somebody—a collector, I think,” Ms. Tompkins said. She recalled him saying firmly, “No, that’s not what I do. I teach school so I can have the privilege of saying no to things like this.’”

After working at Pace, Ms. Greenspon became an art advisor and then worked for dealer Marianne Boesky, where she organized a Krushenick show after Mr. Algus put her in touch with his widow. Her father, William Greenspon, whose extensive art and Bauhaus furniture collection has been profiled in The New York Times, had impressed on her a mission similar to Mr. Algus’s. “‘There are these opportunities, Amy, you should be looking for,’” she said he told her of the underrated Krushenick.

Mr. Algus’s gallery, which he moved to a small second-floor space in Chelsea in 2002, was also an inspiration for her. “I would go over and see his shows and sit down and talk. It was a refreshing and interesting place, a very special environment. Artists were always coming in and out.” She proposed a partnership.

“I had been doing it for so long by myself, it sounded like something that was worth trying,” Mr. Algus explained. “It’s easy for me to do shows, and it’s better if I don’t have to deal with all the other stuff.” Plus he has a lot more space.

Mr. Algus and Ms. Greenspon have had what’s referred to in couple’s therapy as constructive arguments. “Sometimes he would show me things, and I would be like, ‘My God, that is the most hideous thing,’” said Ms. Greenspon. “Then I would think, ‘He’s right. This artist is up to something interesting.’ The same goes in the other direction. Artists of my generation who I found to be really intelligent, if it didn’t look like it was made by a 75-year-old man or woman, it wasn’t really his vibe, but he also comes around.”

She began representing her artist friends Emily Sundblad, Adriana Lara and Mathew Cerletty, who helped design the space. “Because it is much more professional, suddenly artists saw it as something hot,” said Adrian Dannatt, an art journalist and friend of Mr. Algus.

“A more effective way for a gallery to make money is to chase trends,” said Mr. Cerletty, a painter who dated Ms. Greenspon for two years. “But I think it appealed to artists that the gallery’s program has integrity.” Mr. Algus’s gallery always had integrity, but his new place also has buzz.

The two partners don’t always agree about whom to show. “The most difficult thing right now is not having the freedom to be nice to people,” Mr. Algus said. Before, he could show an obscure artist on a whim. “Before, I could be nice to everybody.” Said Ms. Greenspon, “The space demands a certain level of quality and reverence.”

The new space was pricey. “We don’t have a backer,” Ms. Greenspon said. “It was outrageously expensive and it was money we had saved over the years.” As she watched that new floor go in, she said she thought, “How many Murakamis did I have to sell for this?”

At Eli’s, Mr. Algus made the partnership sound not exactly harmonious. “Wait, he went there?” Ms. Greenspon said incredulously, on the phone. “It’s not, for us personally,” she said. “But for the good of the gallery it’s pretty genius—our complementary, sometimes antagonistic approaches. We hadn’t curated or done anything together like that before, and if we had, maybe we wouldn’t have opened the gallery, but I’m glad we didn’t, because it’s forced us to work together and compromise.”

She said she didn’t expect him to be such a “spendthrift.” She was half joking. He started calling himself “the adult” before cutting himself off. But while he’s pained about not showing some of his funkier oldies, he seems to get a kick out of her younger artists. She can now admit that her idea to inaugurate the gallery with a Balthus retrospective was “totally naïve”; then again, she’s been busy selling sculptures from their current show of Algus favorite Bill Bollinger to, she bubbled to The Observer, “really good places.” They seem to find each other amusing.

But can such an unusual alliance endure?

“They definitely butt heads and have fights,” Mr. Cerletty said. “It seems like the gallery is going to close every time there is a show, but it usually works out, and everyone thinks it looks better for having fought it out.”

When we asked Mr. Algus about selling, he looked uncomfortable. “I hate telling people what to think,” he said. “The shows do it. I’ve listened to very well-known dealers do it, and it gives me the creeps. It makes my skin crawl. Why should I have to sell this to you?”

arusseth@observer.com