Varieties of criticism
Conservative criticism has nothing much to do with "left-wing" or "right-wing", because left-wing people are often very conservative, in the defined sense, while right-wing people can also be very radical, in the defined sense. The difference between "radical" and "conservative" has more to do with the belief in whether "a change to something genuinely new" is really possible and necessary. Radicals typically believe strongly that such change is highly desirable and necessary, and that it can be achieved. Their criticism is that there is "not enough" change. Conservative critics, by contrast, are very skeptical about any such change, because they feel the change is really "just another form" of something that already exists.
Council of Conservative Citizens
The CofCC considers itself a traditional conservative group opposing liberals and neo-conservatives; it supports national self-determination, immigration restriction, federalism, and home rule, and opposes free trade and global capitalism. Its specific issues include states' rights, race relations (especially interracial marriage, which it opposes), and conservative Christian values. They have criticized Martin Luther King, Jr., who is considered by the organization as a left-wing agitator of Black American communities with notable ties to communism, and holding personal sexual morals unworthy of a person deserving national recognition. They consider the American Civil Rights Movement and the Frankfurt School as elementally subversive to the separation of powers under the United States Constitution. The Council of Conservative Citizens is active in organizing the restriction, reduction, or moratorium of immigration, enforcing laws and regulations against illegal aliens, ending what they see as racial discrimination against whites through affirmative action and racial quotas, overturning Supreme Court rulings and Congressional Acts such as forced busing and gun control, ending free trade economic policy, and supporting a conservative sexual morality, which includes promotion of the Defense of Marriage Act and opposition to the inclusion of homosexuality as a civil right.