The bright white hats may be the most distinctive feature of the uniforms worn by the 50-person civilian ticket-writing platoon, which for the past two months has made life more miserable for parking violators in downtown Washington.

Yet another eye-actcher-in a different way-is the shoulder patch on those uniforms. Its depicts the dome of the U.S. Capitol.

It is a somewhat peculiar symbol of official for a District government agency established in 1978. Only five years ago, this city capped a century-old struggle by wrenching a form of limited home-rule from the historically tight-fisted powers-that-be beneath very dome.

Symbols are important, some people believe. Last year, for example, City Council member Marion Barry (now the mayor-elect) began a subsequently abandoned effort to do away with "a daily reminder of the colonial rule imposed on us for decades" by changing the name of the District Building to City Hall.

John Brophy, the city public parking administrator, said his agency's intentions were slightly different from Barry's when the parking aides' patch was drawn up. "we discussed a lot of different designs," Brophy said. "The first suggestion we had was a replica of the District Building. But who the hell would recognize it?"

Brophy said the patch had to be immediately identifiable and to look official and authoritative. People had to understand that the person writing the tickets was not a police officer, Brophy said, nor just someone from a private, uniformed guard service and in no way connected with the city government.

In Washington, D.C., things generally considered official-looking tend to be things federal-like the Capitol dome and the Washington Monument, he said.

The District police department patch bears prominent replica of the Capitol dome. The city seal-cluttered with two figures, the capitol, an eagle and other do-dads-is "indecipherable" when made into a patch, Brophy said. The stars and bars of the city flag can be found, for starters, on signs advertising a local bank.

Then, there are abstract originals like the green-and white, swish-lined logo of the D.C. Department of Enviromental Services, which Brophy thinks looks like "a swastika." So, he asked, what is a well-intentioned, homegrown District of Columbia civil-servant-turned-patch-designer to do?

"Maybe it shows a glaring lack of immagination on our part," Brophy said. "A badge is a symbol of authority. If someone could come up with a better symbol that most closely represents the District, given the home-rule status that we have, i'd be happy to reconsider."

Ironically, mayor-elect Barry is now downplaying the significance of symbols, saying through his press sectetary that his administration will concentrate on "substance rather than style."

Still, press secretary Florence Tate said, the city seal and flag will be prominently displayed during Barry's inaugural Jan 29 Once in office, the new administration plans to junk the present mayoral stationery(with a city seal at the top) and replace it with a still-to-be-designed pattern emphasizing "the urban community of Washington."

Ivanhoe Donaldson, mayor-elect Barry's shrewd, low-profiled and ironwilled alter ego, has personal responsibility for finding a way to remove from power the Barry administration acting city administration Julian R. Dugas-who in many respects has been to Mayor Walter E. Washington what Donaldson has been to Barry.

Many of Barry's key supporters consider Dugas a symbol of alleged insensitivity in the Washington reinstated Dugas as director of the D.C. Department of Licenses, Inspections and Investigations and reappointed him chairman for the next three years of the influential Alcoholic Beverage Control Board.

During his campaign, Barry pledged to fire Dugas as city administrator if elected. That was somewhat meaninless promise, since the city administrator serves at the pleasure of the mayor,anyway.

In his newly regained job (dugas held the same post before becoming city administrator in 1975), however, Dugas has civil service protection against dismissal without due cause. What Dugas wants, some friends say, is to stay in city government for at least several monthss-and perhaps a couple of years-in order to obtain better pension, benefits when he retires.

The two men, Donaldson and Dugas, could perhaps not be more perfectly matched. Both are feisty, at times surly and stubborn, at their best in situations that parallel hand-to-hand combat and deceptively rugged and cold-blooded.

Donaldson, a wiry 155-pounder, has been known on at least one occasion to barroom brawl with someone approaching twice his size. Dugas is 60, a no-slouch lawyer by training and takes nothing more serious than what he considers his "own personal business."

The key to politically more peaceful resolution could rest with an intermediary well-known to both men. One such person, an observer says, is Herbert O. Reid, former dean of Howard University Law School (Dugas' alma mater) and a confidant of Barry.

Otherwise, the observer said last week, "I'll take Julian to win the first round. But after that, I wont't call it."