In fact, there are good reasons to think that Clinton and Ryan might be able to reach compromises if she is elected. There are at least five major areas in which she and the speaker could find common ground.
Reforming public assistance has always been one of Ryan's main objectives in public life, and he has produced detailed proposals on how he thinks the government can better help Americans in need. Ryan thinks that, as currently structured, programs such as food stamps and housing vouchers discourage beneficiaries from working, making them dependent on public assistance, and he thinks that the goal should be to encourage them to find employment.
To be sure, Clinton would likely reject some of Ryan's ideas, such as taking the money used to fund federal assistance programs and turning it over to state officials to make decisions about how it should be used. Yet Clinton does share with Ryan the fundamental philosophy that work is an important part of these policies.
Clinton stands by the law her husband signed in 1996 when he was president that overhauled welfare and required poor Americans to work or participate in training or other employment programs to receive cash from the federal government
Although she supports some changes to the law, such as eliminating the five-year lifetime limit on benefits for any one recipient, she has defended the legislation's basic principle. "Bill and I, along with members of Congress who wanted productive reform, believed that people able to work should work," she wrote in 2003.
A statement earlier this year from Maya Harris, one of Clinton's advisers, praised the law's measures to promote employment. "It was done with a package of reforms like expanding the EITC, job training, and childcare, so people would have the tools they needed to find work and take care of their families," Harris said, using the acronym for the Earned Income Tax Credit, which is a federal bonus designed to encourage poor Americans to work.
Meanwhile, researchers in both parties have been developing other ideas to support work that Clinton could endorse -- such as an increased minimum wage combined with a subsidy for employers, or relocation credits to help people move their families out of depressed regions and find work elsewhere.
"You could have a robust bipartisan agenda to deal with both poverty and the working poor," said Norman Ornstein, a resident scholar at the American Enterprise Institute. "You have a very significant amount of Democratic support, and something I think Clinton would be eager to get behind."
2. Criminal justice
Part of the plan for poverty Ryan published in 2014 included a chapter on sentencing reform. The document called for reductions in sentences for nonviolent offenders, noting that incarceration damages a person's economic prospects.
Clinton's first major speech of her campaign, more than a year ago, was on criminal justice. It seems that both politicians are eager to act on the issue, and a sentencing-reform bill in the House has won praise from conservative reformers as well as liberal activists.
Presumptive Republican presidential nominee Donald Trump, by contrast, has taken a tough-on-crime stance, saying that protesters in the Black Lives Matter movement are "looking for trouble" and calling police "the most mistreated people in this country."
Clinton has said she opposes President Obama's international trade deal, the Trans-Pacific Partnership, which she helped negotiate as secretary of state. Ryan supports the deal -- he has called it "very important" for U.S. influence in the global economy.
Clinton's ambivalent comments about trade, however, have been much more favorable than Trump's protectionist invective on the stump. She has not ruled out the possibility of pursuing additional agreements if she is elected president.
"I did say, when I was secretary of state three years ago, that I hoped it would be the gold standard," Clinton said of the Trans-Pacific Partnership at a debate in October. "It was just finally negotiated last week, and in looking at it, it didn't meet my standards, my standards for more new, good jobs for Americans, for raising wages for Americans."
Especially in contrast to Trump's more forceful rhetoric, careful remarks such as that one suggest that if future deals meet Clinton's standards, she and Ryan might be able to work together on international trade.
Another long-standing goal for centrists in both parties has been a major federal investment in highways, railroads, ports, water systems and other physical infrastructure.
"There are a lot of Republicans who are ready to put some shovels into the earth," said John Hudak, a political scientist at the Brookings Institution.
Ryan was reportedly working on bipartisan infrastructure legislation this fall. Clinton's campaign has said that if elected, she would make infrastructure an immediate priority, asking Congress for at least $275 billion. So far, Congress hasn't approved new spending on a large scale, but that might change if Clinton becomes president.
To be sure, a President Trump might devote time to negotiating an infrastructure bill, too. Trump has also talked about the need for investment during the campaign -- during a debate on foreign policy, he digressed to talk about how the money spent on the war in Iraq could have been better used.
"We've spent $4 trillion trying to topple various people," he said. "If we could've spent that $4 trillion in the United States to fix our roads, our bridges and all of the other problems -- our airports and all of the other problems we've had -- we would've been a lot better off."
5. Foreign policy
Ryan has not focused on foreign policy to the same degree as he has talked about domestic issues, but his column in the Gazette did promise that his caucus would be advancing some kind of document on international affairs in the coming weeks.
"We’ll present the ideal national security and foreign policy to keep Americans safe," he wrote. Presumably, whatever strategy he and his colleagues present will reflect the hawkish inclinations of many House Republicans, which are at odds with the isolationism of their party's presumptive nominee.
Clinton will have her differences with GOP hawks. She has said she supports Obama's nuclear accord with Iran, for example. Yet she has been more willing to use military force than her former boss -- she favors military intervention in Syria -- and it seems that she and Republican lawmakers could work together on foreign affairs. Already, some leading GOP experts on foreign policy have said they're supporting her, including Max Boot, who has advised Sen. Marco Rubio (R-Fla.).