Washington's Cardozo High School band, which must be almost everybody's favorite high-stepping musical outfit, will have its first chance next month to show its stuff at a place about 20 blocks from its campus: The White House.

Cardozo's band will appear June 12 under the honorary patronage of President and Mrs. Reagan at a picnic and reception celebrating what is called the presidential premiere of the movie "Superman III" for the benefit of the Special Olympics. Film stars Christopher Reeve and Jackie Cooper will attend.

It will be the band's first appearance at the presidential mansion, according to George Gill, the band director, who noted that his unit performed outside the home of Vice President Bush for a similar benefit premiere for "Superman II" in 1981. The band also marched as the capital city's proud representative in the 1981 Tournament of Roses parade in Pasadena, Calif.

The Special Olympics, needless to say, is the world's largest program of athletic competition for mentally handicapped persons. It's thoroughly nonpolitical: Eunice Kennedy Shriver, sister of President John F. Kennedy, is its founder and president.

Some tickets, costing $250, are still available for the movie premiere and White House reception, according to the sponsors. For $500, they'll throw in lunch with the stars at the Shriver home. For information, call 331-1346, or write Special Olympics, 1701 K St. NW, Suite 103, Washington, D.C., 20006.