We Americans have a new enemy, and it is ourselves. Half of Democrats and Republicans now see members of the opposing party as not just ill-informed but actually frightening, according to the Pew Research Center. In experiments with thousands of Americans, researchers have found that partisan animosity can now exceed racial hostility: When awarding hypothetical scholarships and cash, we are more likely to discriminate based on politics than race. As of 2010, a third of Democrats and half of Republicans said that they would be upset if their children married someone from the other party—up from about 5% from each group in 1960.

Why do we hate each other so? One reason is that Americans are more politically segregated than in the past. We are less likely to have neighbors who belong to another party than we were 20 years ago. And one thing we know about human behavior is that it is easier to hate people you don’t know.

Seventy years ago, almost to the day, a group of American ambulance drivers, disgusted by the waste and carnage they’d seen in the World Wars in Europe, started up a cultural exchange program between Europe and the U.S. The idea was simple: If people knew each other—really knew each other—they would be more tolerant of one other. It’s harder to demonize someone once you’ve stayed in their homes and shared meals and stories together.

So after helping to evacuate Germany’s Bergen-Belsen concentration camp, the American Field Service ended its volunteer ambulance corps and became a student exchange program. In 1947, 53 students from 11 nationalities came to the U.S. and France to study and live for one year among foreigners.

“You can’t know each other and be enemies between yourselves,” President Dwight Eisenhower told a group of 1,150 foreign exchange students in Washington on the 12th anniversary of AFS. Since then, the global program has brought hundreds of thousands of young people into homes and schools far outside their comfort zones, building trust and relationships across oceans.

Then, in the 1970s, the U.S. chapter of AFS started an additional program for exchanges within the country. Over the life of the program, more than a thousand teenagers would travel from Kansas to California, from Florida to New York and beyond—crisscrossing the country to live for a summer or a semester with host families, discovering new food and strange customs within their own country.

In June 1979, 17-year-old David Angel took a Greyhound bus from Goodland, Kan., to McConnellsville, N.Y. As his bus pulled into the station, his heart pounded in his chest. He would be living in a very small town in upstate New York, near Syracuse, but in his mind, it might as well have been Tokyo. “I imagined people from New York being very sophisticated and continental,” Mr. Angel said, “and I was afraid I’d be Mr. Redneck Country Bumpkin.”

But his host family was warm and welcoming: “They met me at the bus station and just wrapped me up in their arms.” That summer, Mr. Angel mixed concrete, swam in rivers and went fishing. Each evening, his host parents, grandfather and sister would gather in the living room for drinks and conversation, and Mr. Angel would listen and learn, sometimes chiming in with his own ideas. To cap off the experience, he spent a wild two weeks in New York City, visiting the Twin Towers and the Statue of Liberty and seeing “The Rocky Horror Picture Show.”

The experience left him more culturally agile. “I’m in my 50s now,” he says, “and even to this day, I use that tool regularly—to remind myself, ‘OK, this is my opinion, and that is their opinion, and there are different opinions.’ It makes everything less black and white.”

Today, Mr. Angel lives in Los Angeles, and some of his friends lack this skill, he says. “Either you’re a Republican, or you’re dead to them. And some of my Democratic peers are the same way,” he says. “I find that troubling. There is so much that our country has to offer.”

AFS-USA’s domestic-exchange program ended sometime in the mid-1980s, as the organization returned its focus to international exchanges only. But since the 2016 election, its leaders have heard from people who want to restart the domestic program.

“These days, we are asking how we can contribute to building a more just and peaceful world—right here in our own country,” says Jorge Castro, the president of AFS-USA, based in New York. The organization is sponsoring a national survey later this year to gauge potential interest in such a program and beginning to contemplate ways to fund domestic exchanges. “Before knowing or understanding other cultures abroad, you have to understand your own,” says Mr. Castro.

In 2011, as part of a book project, I followed three American exchange students who were spending a year living and studying in countries far from their homes. At first, I was struck by their courage in deciding to go to South Korea or Finland, when it would have been much easier to stay home in Minnesota or Oklahoma, surrounded by the familiar and the comfortable.

But over time, I saw something even more impressive. When I visited these teenagers in their borrowed cities and villages, I could tell they were becoming more flexible, more at peace with uncertainty. Yes, they had developed empathy for other points of view, which is priceless. But they were also letting go of some of the fears that hold so many of us back. It was like they had shrugged off a layer of assumptions and judgments and could move more easily through the world, unencumbered.

Americans relied on domestic-exchange programs at another point in our history: in the 1960s, a time of wrenching tensions over war and race. Back then, a group of historically black colleges and universities set up domestic exchanges with majority white universities in other states. And just as AFS did after World War II, the exchanges forced interactions that would never have happened otherwise.

Many of those exchanges faded away with time, but not all. Every year, Brown University in Providence, R.I., and Tougaloo College, a historically black college near Jackson, Miss., still host exchanges—connecting a blue state to a red one, not with commerce or webinars but with human beings who arrive with duffel bags and stay past the point of discomfort. More than 500 students and faculty have participated in the exchange over the past 50 years.

“It’s almost like two different countries,” says Waynesha Blaylock, 23, who grew up in Texas and Mississippi before spending a semester at Brown in the spring of 2015. During her first weeks in Rhode Island, what she missed most was the Southern hospitality—the way her adviser at Tougaloo would welcome her to her office every day; the way everyone would say hello on campus, without fail. At Brown, most people seemed cold and unwelcoming. And she ached for a good Sunday dinner. “I could not find any good fried chicken. I was really heartbroken about that,” she told me, laughing. Also, “sweet tea” in Rhode Island is not really sweet tea, she discovered.

But over time, she learned to reinterpret that aloofness—to see it as cultural, not personal. And she taught the Brown students too, giving a talk in her public-speaking class on the value of Southern hospitality, for example. “Most people in that class, they would all wave and stop and talk to me after that,” she remembers.

Today, Ms. Blaylock is in medical school—back at Brown University. She wouldn’t have thought to apply there if not for the exchange program, she says: “I didn’t know what an Ivy League school was.” But she still finds herself playing the role of anthropologist every so often. “There are plenty of people here who have no idea what the South is like,” she says. And that ignorance can have real consequences for physicians in training. “We have discussions about obesity,” she said, “and I have to explain that it’s just not that easy to change your diet in the South.”

The Brown students who come to Tougaloo have their own struggles. “We had a Brown student who had spent a semester in Amsterdam, and she said it was nowhere near as different from Providence as Tougaloo,” says Elise Morse-Gagné, who teaches English at Tougaloo. “The food is very different. The economy is different. The attitude toward education is different. Religion is colossal.”

She herself came to Tougaloo from New England. Over the years, she’s learned to recognize biases in both places. In the North, for example, “white liberal academics tend to be extremely dismissive of Christianity,” she says. In Tougaloo, that kind of prejudice would be problematic. “Every meeting on campus here starts with a prayer, and it’s a Christian prayer, and it’s just taken for granted,” she says. “You can’t afford to be dismissive of the beliefs of the people you live around.”

Two years ago, Ms. Morse-Gagné’s husband died, ahead of his time. And again, she was reminded of the cultural differences. “People were warm, they were loving, and they were helpful in practical ways,” she says. They brought food and they took over her classes. The college president and the provost came to the hospital. At a New England university, she suspects, many people wouldn’t have known what to say or do, and the response would have been more strained.

The world is a big, complicated place, but so is this country. Maybe it is time to start treating America as a collection of different countries, one that has lost its common language but remains inextricably intertwined.

“You can’t say that the United States is one culture. We are definitely different in every single community,” says Debra Sanborn, the president of the National Student Exchange—another organization started in the turbulence of the 1960s, which now works with 2,000 students at 160 different North American campuses each year. Many students leave seeking a change of weather or different classes; they don’t go in search of cultural difference, but they often find it.

Even within states, our differences can be bewildering, as Savannah Barrett learned in Kentucky. She grew up on a seventh-generation tobacco farm before moving to Louisville. When she first arrived, she found that city people judged her by her strong country accent. “I’ve been told that folks just don’t think I sound smart,” she says. She and a friend with a similar background were struck by how many biases city people had about country folks—and vice versa.

In 2014, Ms. Barrett and her friend started the Kentucky Rural-Urban Exchange to bring rural and urban people together for immersive experiences three weekends a year. To date, 160 people from 34 Kentucky counties have participated. People of all ages apply to the program, which is free, and about half are accepted, based on the mix of experiences and interests they bring to the group. The itinerary is carefully curated to deepen participants’ understanding of Kentucky’s history, culture and identity, while also equipping them to solve specific community problems.

But it’s essential to make people comfortable, too, Ms. Barrett has learned. That’s when they let down their guard. “We start and end every weekend with a meal,” she says. The program reflects the identity of the different Kentucky towns in which it takes place and typically includes a hike, a cultural performance and a visit to a swimming hole. “What we have found is that it’s really difficult to hate someone you’ve had deep, meaningful, transformative experiences with—and whose community you’ve really come to value because you’ve felt welcome and appreciated there.”

This holiday weekend, Ms. Barrett will facilitate her own micro-exchange: bringing a group of her Louisville friends to the farm where she grew up. That’s her July Fourth tradition. From experience, she knows there may be some difficult moments, but it will be worthwhile. “We crave a deeper connection with folks who are different from ourselves,” she says.

In recent months, people from Minnesota and other states have reached out to Ms. Barrett for advice on setting up similar exchanges, and she is working on creating a shareable tool kit.

We need to make America whole again. The country has been weakened by our mutual disdain and shared ignorance. We should tap the wisdom of Ms. Barrett, AFS and other exchange experts to help us create opportunities for Independence Day exchanges across the country—an annual tradition where Americans open up their backyard barbecues and remember what we have in common.

We don’t have to agree about politics (we won’t); we don’t even have to talk about Donald Trump (probably better not to). But as Americans, we share very few cultural traditions. So we should make July Fourth a time to come together and get reacquainted. We share more than our affection for grilled food and fireworks; we also believe in the right to life, liberty and the pursuit of happiness.

And if that isn’t enough, we Americans also like to win. Right now, a majority of Democrats and Republicans say members of the other party are “closed-minded,” according to a 2016 Pew survey. Wouldn’t it be fun to prove each other wrong?