When pressed to expand on their relationship, though, Banga insists that Haythornthwaite acts chiefly as a facilitator between him and the board — not as his ally.

You can’t blame Banga and Haythornthwaite for downplaying their collaboration. A nonexecutive chairman, after all, is meant to be an independent, not the CEO’s chum. But both leaders, in principle, should be working toward some of the same goals: looking out for the interests of the company’s shareholders, ensuring the business is well managed, and soliciting the advice and feedback of other corporate directors.

Striking the right balance of collegiality and autonomy isn’t easy. Indeed, many companies don’t even try; at more than half of big U.S. corporations the CEO and chairman are the same person, according to research firm Equilar. And investors seem fine with this: In mid-May, for example, J.P. Morgan Chase (JPM) and Pepsi shareholders voted down proposals that would have stripped those companies’ CEOs of their chairman roles.

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As part of Fortune’s ongoing Executive Dream Team series, which will look at successful alliances in the boardroom and C-suite, we asked Banga and Haythornthwaite to offer a glimpse of how they navigate the delicate CEO-chairman relationship. Their answer? Carefully.

Haythornthwaite’s approach is a variation of the axiom “Trust, but verify.” A seasoned director who also serves as nonexecutive chair of Network Rail, a private company that manages railroad infrastructure in Britain, he readily acknowledges that a nonexecutive chair doesn’t insulate a company or a board from bad decisions. In fact, independent chairs are standard in Britain, but they didn’t prevent corporate crises at BP (BP), for example. “Ajay knows I will back him to the hilt — unless and until I don’t.”

Banga in turn looks to Haythornthwaite for a range of insights — soliciting his views, for example, before accepting a new role as chairman of the U.S.-India Business Council. But make no mistake: Banga doesn’t expect Haythornthwaite to get into the day-to-day operations at MasterCard (MA), based in Purchase, N.Y. Instead he looks to him for big-picture advice. “In Rick and the rest of the board, I get a sounding board of 10 to 11 really bright people who get paid relatively little to offer me their wisdom,” says Banga.

Many corporate watchdogs and executives feel strongly that other companies should follow MasterCard’s lead and split the CEO and chairman roles, but studies show that companies that insist on separate roles don’t necessarily outperform those that embrace the CEO-chair model. To wit: MasterCard shares are up 48% in the past 12 months; shares of Visa (V), where Joseph Saunders serves as chairman and CEO, are up 45%. “Ajay and Rick work really well, and I salute them, but it’s not a panacea,” says Jeffrey Sonnenfeld of the Yale School of Management. “It’s just one of many ways of governing a complex enterprise.”

Perhaps the best way a chairman, nonexecutive or otherwise, can help a company is by fostering an independent and lively board of directors. If the board is truly engaged, insightful, global, and diverse in background, it is hard for a CEO to grab too much power or reject the counsel of his or her advisers.

And a strong board can prevent a nonexecutive chairman from falling under the sway of the CEO, a risk that Haythornthwaite and Banga take great pains to avoid. “The relationship between Ajay and me is vital. The relationship between the two of us and the whole board is even more vital,” Haythornthwaite says. “And nothing is more divisive than something that appears to be a kitchen cabinet.”

This story is from the June 11, 2012 issue of Fortune.