Female counterpart? We now have a male Republican Party and a female Republican Party?
Long-standing tradition that both sexes have to be represented in party leadership. I think it goes back to the suffrage movement. At the precinct, county, district and state levels, the Chair and Vice-Chair must be of opposite sex. At the national level, the Chair and Co-Chair must be of opposite sex.
Each state also has a National Committeeman and a National Committeewoman.