Whatever President Reagan sends to the new Congress next year, including his proposals for reducing the federal budget deficits, he again will have to deal with a contentious Democratic-controlled House of Representatives. It's the old checks and balances at work.

Ah, but things are never as tidy in real life as they could be. In a perverse sort of way, the political fortunes of the Republican president and Democratic congressmen have become entwined. They ought to sit down and buy each other drinks.

It began after the 1982 elections, when Speaker Thomas P. O'Neill and the Democratic House leadership regained their working ideological control of the House.

In 1983 and 1984 they headed Reagan off on the MX missile and funding of military and paramilitary operations in CentralAmerica. They forced him to extend the Voting Rights Act, and they put restraints on proposed cuts in a host of domestic programs.

For Reagan, it was a dramatic change from the heady days of 1981, when the combination of his big victory at the polls, Republican control of the Senate and a conservative coalition of Republicans and Boll Weevil Democrats in the House enabled him to ram much of his program through Congress.

But you never know how these things are going to work out.

What in fact was the result of the Democrats' valiant opposition of the past two years? The Democrats helped prevent, in their view, inestimable damage to the republic. But some are wondering what else they wrought. They suspect that they may have been among Reagan's most potent allies in his landslide reelection this year. Ironically, their private view is that he owes them a big one.

"Letting Reagan be Reagan, which would have let him cut Social Security as much as he proposed back in '81, get the MX and fight a secret war in Nicaragua, would have been a disaster for him," says one leadership aide. "In a sense, we've housebroken Reagan. We're like a guy's friends who hold him back from a barroom fight and keep him from getting his face pounded. We sanded off the rough edges. We made Reagan a conservative rather than a radical."

This is the Founding Fathers' checks-and-balances system working before our very eyes, even though it doesn't always work quite as they intended. While there is no evidence that the voters cast their ballots with the idea of establishing checks and balances, there are indications that this is what they want.

Every recent poll that has had a question on the subject showed that the American people approve of having one party in control of Congress and the other in the White House to keep a suspicious eye on each other.

The polls indicate that the people like having Reagan in the White House to counter the big-spending liberal Democrats. By the same token, a recent Harris Poll showed that 60 percent liked having the Democrats on the Hill as a check on Reagan.

The Democrats' gain of 26 House seats in the 1982 elections, which was generally credited to the economic recession, restored their check on the White House.

A lot of Republicans understandably hoped that Reagan might carry enough Republicans in with him in his landslide to offset these losses and give them a reprise of the Golden Age of '81-'82. Now they are complaining that his Feel-Good campaign wound up benefiting Democratic as well as GOP incumbents, to the detriment of Republican challengers.

But they may be missing the point. The real villains responsible for their disappointment are those sneaky Democrats.

Walter Mondale may have seen the need for a tax increase, but many Democratic House candidates saw that Reagan was opposed to it. They also saw that Reagan was very, very popular. They remembered that one of the basic rules of politics is never raise taxes in an election year.

They didn't have to have a roof fall in on them. A lot of them took a campaign stand at least as sturdily opposed to a tax increase as Reagan's.

A lot of them were reelected. And people who don't want their taxes raised next year can relax and not worry about the system of checks and balances, because in this particular case it may well not be working next year. When it comes to holding the line on taxes, Ronald Reagan may have more support on Capitol Hill than people who don't read past the election returns realize.