The D.C. City Council, in its fourth legislative session since home rule, has gotten off to a rancorous start. But the first big battle of 1981 is not over such past favorite issues as workemen's compensation, rent control or the city budget.

This time the dominant fight is over office space and staff for the 13 council members.

The feud has split the council into two warring factions, with Chairman Arrington Dixon and four allies on one side and at least six members on the other. The first floor of the District Building, where the council members have their offices, was still littered yesterday with the debris from the combat -- desks and office furniture stacked high in the corridors. As a result, committee meetings on proposed legislation have yet to be scheduled.

Meanwhile, the council members are fiercely waging their private dispute through a series of memos and countermemos they dispatched from office to office.

Council members are accusing Dixon of rewarding the members who voted with him on the reorganization of the council at the opening meeting last Friday by giving them their choice of office space and extra staff.

"What he's doing is getting even with all of us," complained council member Nadine P. Winter (D-Ward 6), the chairman pro tempore, who voted against Dixon on Friday. "The issues are so much greater than office space and staff. What I'm concerned about is the attitude of the leadership."

Dixon informed Winter after the Friday meeting that he was taking away one of her staff members, a clerk who was assigned to her from Dixon's Committee of the Whole staff. Dixon's press spokeswoman Carol Richards said he made the move because Winter, as chairman pro tempore, will no longer have as many duties in hearings on confirming nominees for city jobs under the newly adopted internal council rules.

But Winter is complaining that the staff member is being transfered to the committee staff of Councilwoman Wilhelmina Rolark (D-Ward 8), who voted with Dixon, because Winter voted against the chairman's reorganization plan. Winter is appealing Dixon's decision to D.C. Auditor Matthew Watson, accordng to a memo leaked to the press.

According to another memo made available to the press, Dixon has assigned an extra committee staff member to council member William Spaulding (D-Ward 5), another political ally who voted with Dixon last Friday, in a move Dixon critics claim was politically motivated.

Meanwhile, some council members are steaming over the fact that Dixon unilaterally assigned council member Jerry A. Moore Jr. (R-at-large) to the corner office left vacant when former council member Willia J. Hardy retired. The corner offices on the council floor are larger than most, with a better view, and provide more space for a committee staff. Moore voted with the chairman.

But before Moore could move into his new office, he had to wait until Councilwoman Charlene Drew Jarvis (D-Ward 4) moved to another office. Jarvis had moved in on New Year's Day, when no one else was around, to establish what another council member called "squatter's rights."

Council member David A. Clarke (D-Ward 1), who voted against the chairman's reorganization plant, accused Dixon in an internal memo of reneging on a promise to hold a meeting on a promise to hold a meeting to allocate office space. Clarke said "rumors are rife within the council as to alleged actions that you are taking or have taken based assertedly upon your discontent with certain members' votes at the legislative session on Jan. 2, 1981." If the rumors are true, Clarke told Dixon, it would "serve you little and could possibly lead to the factionlization of the council."

Richards said yesterday that Dixon made the office space assignments unilaterally because "the council chairman is the chief executive officer of the council." According to the home rule charter, Richards said, "he's not really overstepping his bounds."

The battle lines were drawn after the coucil's heated opening meeting on Dixon's controversial reorganization plan. Dixon wanted to elevate Jarvis, his political colleague, to the chairmanship of the council's powerful Housing and Economic Development Committee, while at the same time passing over two more senior at-large Democrats, John Ray and Betty Ann Kane, who wanted the same post.

Such chairmanship squabbles are usually settled in private caususes, and only formally ratified at the council's opening meeting. But this year, the behind-the-scenes bickering spilled into the open, giving the public a first-handed glimpse of the deep divisions on the council.

The whole episode has left an atmosphere of confusion at the District Building. "I don't thrive on confusion," Winter said. "I get very confused by confusion. I just hope the newspapers don't pick up on this."