The willingness of House Democratic leaders to compromise on a deadlocked issue late in the just-completed session after a threat to investigate Speaker Jim Wright leads Republicans toward a prickly new strategy.

At issue was a Democratic effort to keep open the Iran-contra investigation through election-year 1988. Another of those party-line House votes overwhelming the Republicans was in prospect. But the strident atmosphere abruptly changed to mellow compromise for closing the probe.

Was it because Rep. Newt Gingrich had filed a one-minute speech proposing a vote on a bipartisan ''special ethics'' committee ''to investigate allegations of improper and unethical conduct'' by Wright? Democratic leaders laugh that off. But whether the idea is laughable or not, a great many Republicans think Gingrich got the Democrats' attention.

That in itself changes the mood in the House. If an iron grip there since the 1982 election has made Democrats arrogant, Republicans have grown desperate. The session-ending events have made them prone to follow Gingrich's game plan of charging personal corruption by House Democrats in general and Wright in particular.

Not many Republican colleagues were dancing months ago when Gingrich, a resident GOP gadfly and one of the chamber's few political innovators, sat down at the piano to play his tune. Although Wright is considerably less popular than Tip O'Neill, the old Republican bulls prefer civility in dealing with Democratic counterparts. Even Gingrich's fellow bomb-throwers on the back bench felt the corruption issue was too exotic.

Much of this skepticism melted away during backstage events Dec. 9 and 10. At that time, Rep. Dick Cheney, House GOP Conference chairman, had run into a stone wall trying to negotiate the close of the deeply partisan Iran-contra probe. Cheney wanted the doors shut and the files sealed Feb. 15. The Democratic position, pressed by House Majority Leader Thomas S. Foley, would have kept the select committee going all year, with staffers having open access to the files.

On the evening of Dec. 9, Cheney was informed that the House Rules Committee had just approved the undiluted Democratic position. The Republican alternative was drafted with justifiable expectation that it would be summarily crushed by party-line discipline. It was then that Gingrich told Cheney he might as well make things interesting.

The next morning Gingrich filed a proposed one-minute speech citing a Washington Post investigative report of how Wright received 55 percent book royalties from a friend whose printing company worked for his campaign, plus a Regardie's magazine expose about Wright (''The Speaker and the Sleaze''). Contending those articles ''combine such a massive list of allegations of misconduct and unethical behavior that we clearly have an obligation to investigate them,'' he proposed a special committee.

A few hours later the Democrats turned more reasonable on ending the Iran-contra investigation. It would terminate March 1, with the files sealed for members and staffers unless specifically authorized by the House Intelligence Committee. The Dec. 10 Congressional Record exudes bipartisan sweetness and light in approving this settlement. Gingrich's one-minute speech was not delivered.

House Democratic leaders categorically deny any connection between Gingrich and the compromise. Majority Whip Tony Coelho says the Rules Committee proposal was merely a backup in case no compromise was reached. Foley says the compromise was offered ''when I saw how upset {Minority Leader Bob} Michel and Cheney were,'' adding that the speaker was not consulted.

But many Republicans, while acknowledging they cannot prove it, believe Gingrich's threat did the trick. They include Cheney, never a member of the Newt Gingrich fan club and on the opposite side from him in House factional politics. Democrats point out that Gingrich a few days later assailed Wright in a press conference anyway, but it went largely unnoticed and is not the same as setting off floor debate on the speaker's ethics.

If Republicans follow Gingrich's line, Democrats warn, restraints of civility will be severed. ''If they go after Wright,'' one party leader told us, ''what's to stop us from going after Michel and Cheney?'' The answer from Republicans: What have you got that suggests any financial impropriety by our leaders?

The frustration of House Republicans, particularly junior conservative members, is rising. They see no leadership from a lame-duck White House and no successor to Rep. Trent Lott, running for the Senate from Mississippi, as hard-nosed leader of the right. Facing another year of torment from the Democratic majority, they perceive Gingrich's assault on the speaker as a way out.