WASHINGTON — Rare is the Tea Party-tested Republican senator who hangs an image of the Kennedys’ Hyannisport home over his desk and shows off the painter’s personal inscription.

‘‘Orrin,’’ reads the note, scribbled below a cobalt-blue sea. ‘‘We’ll leave the light on at the compound for you anytime. Ted Kennedy, ’91.’’

The beacon of bipartisanship that defined their odd-couple relationship still guides Utah’s Orrin Hatch. He didn’t advertise it as he wooed and won over the tea partiers who, two years ago, toppled fellow conservative Robert Bennett from his Senate perch.

But now, with his state’s Senate GOP nomination in hand and reelection to a seventh term all but assured, Hatch, 78, can think about his legacy.

He’s very clear about what he wants: As the senior Republican in the Senate and his party’s top voice on the tax-writing Finance Committee, Hatch wants a deal that restructures the tax code while also slowing and even stopping the government’s accumulation of debt. To get it, he says, he will practice the art of compromise over the take-my-marbles-and-leave mentality that has tied up Congress in recent years.

‘‘There has to be a course correction,’’ Hatch said in a recent interview. ‘‘If I am chairman of the Finance Committee, you can bet your sweet bippy I will take a leading role.’’

A new tax code, he says, would have to be bipartisan to pass Congress and, as importantly, have credibility with the Americans who will fork over large chunks of their paychecks under it. Orchestrating it will require a delicate touch with Washington’s most muscular interest groups and stubborn factions of both parties.

In the vaulted Capitol hideaway office he inherited from Kennedy — the seascape hanging nearby — Hatch offered a reality check on how lawmaking happens.

‘‘Neither side is going to get everything they want,’’ he said. ‘‘But it is important that we move ahead, and that we do the art of the doable to pull this country out of the fiscal morass it’s in. And I think we can.’’

Washington’s tribal chiefs know that the conservatism Hatch has emphasized in his reelection campaign coexists with an interest in getting results on Capitol Hill and a long-demonstrated willingness to compromise.

The type of real, red-faced debate that delighted Hatch and Kennedy also produced landmark laws such as the American Disabilities Act and children’s health insurance. Tense talks with no less a partisan than Representative Henry Waxman, Democrat of California, produced a patent exemption that cleared the way for the generic drug industry.

Tax reform could well be Hatch’s enduring legacy. The contours of the debate are clear and broadly philosophical: Republicans think that the government levies enough taxes already but that growing the economy would produce more revenue. Democrats say the wealthiest are not taxed enough.

Much, of course, depends on who wins the White House and control of Congress.

Here is where the debate would start: Presumptive Republican presidential nominee Mitt Romney agrees with Hatch that there should be a one-year extension on all of the Bush-era tax cuts, then comprehensive tax reform. President Obama wants to let those tax cuts expire for Americans making more than $250,000 a year, and then do reform.

No one suggests that Hatch, for all his red meat bluster lately, comes back to the Senate next year any less of a deal-maker. Right now, longtime colleagues say, Hatch is doing what Hatch does best: adapt to the ‘‘rhythms of change.’’