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.
List of left communists
Left communism is a form of communism with its origins in the left-wing of several parties of the Communist International. Its two main original branches were in council communism and Bordigism. This is a list of theorists and political figures who have identified themselves as left communist, communist left, Bordigist or council communist.