The Reagan administration thought it had muzzled the Equal Employment Opportunity Commission earlier this month when White House and Justice Department officials squelched the agency's participation in a controversial employment discrimination case in New Orleans.

But it turns out that the agency's voice still will be heard in the case--and it is still harshly critical of the administration's civil rights policies.

After learning April 7 that the EEOC had been forced to withdraw its "friend of the court" brief in the case, lawyers at two private legal foundations in Washington decided to submit the agency's full brief themselves.

They did it by attaching it to their own "friend of the court" brief, which they then submitted as recommended reading to the judges of the 5th U.S. Circuit Court of Appeals, which is considering the case. Before the other side knew what hit them, the court had officially accepted the submission.

The case is a challenge to a 50 percent racial quota for promotions within the New Orleans Police Department. The city agreed to the quota in the settlement of a discrimination suit brought by blacks in 1973.

Groups of white, Hispanic and female officers, joined by the Justice Department, have attacked the plan as unconstitutional. The EEOC brief said Justice's position had no legal merit and suggested that if accepted it could undermine years of progress in remedying discrimination in the United States.

The legal foundations that relayed the EEOC's message are the William O. Douglas Inquiry into the State of Individual Freedom, affiliated with the American University law school, and the Center for National Policy Review at the Catholic University Law School.

Sidney M. Bach, a New Orleans lawyer representing the white officers, described the maneuver as "sneaky."

But Herman Schwartz, director of the Douglas foundation, said, "Nothing we did was surreptitious. We made it very clear we were adopting the views of the EEOC and those views are the same as ours. Nothing could have been more open."

Bach has asked the appeals court to throw out Schwartz' brief as a "Trojan Horse." Sources said the Justice Department was staying out of the dispute so as not to "dignify" it.