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.
Political correctness
"Political correctness" is a label typically used to describe liberal terms and actions, but not for equivalent attempts to mold language and behavior on the right. However, the term "right-wing political correctness" is sometimes applied by commentators, especially when drawing parallels: in 1995, one author used the term "conservative correctness" arguing, in relation to higher education, that "critics of political correctness show a curious blindness when it comes to examples of conservative correctness. Most often, the case is entirely ignored or censorship of the Left is justified as a positive virtue. [...] A balanced perspective was lost, and everyone missed the fact that people on all sides were sometimes censored."