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.
Rally of Republican Lefts
They defined themselves as "left-wing republicans" whilst they opposed left-wing policies. Indeed, until the end of the 19th century, the French left was defined as republican and the right as pro-monarchy. Then, when the republic was no longer questioned, the conservative republican groups, who had sat at the center-left of the assemblies, moved to the right-wing seats, but they continued to consider themselves as left-wingers: this is known as "sinistrisme".