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.
Left communism
Left communism is the range of Marxist viewpoints held by the left-wing of communism which criticizes the political ideas and practices espoused—particularly following the series of revolutions which brought World War I to an end—by Bolsheviks and by social democrats. Left communists assert positions which they regard as more authentically Marxist and proletarian than the views of Marxism–Leninism espoused by the Communist International after its first congress (March 1919) and during its second congress (July–August 1920).