Importance weighted aggregation plays a central role in utilization of information resources for information retrieval and fusion, pattern and object recognition, decision making, etc. A class of aggregation operators of particular interest is formed by the aggregation operators between the min (minimum) and the max (maximum), the so-called averaging operators. Two key issues in the choice of such an operator for a given application are the kind of importance weighting and the andness (“minness”) of the operator. Two main kinds of importance weighting for such operators, namely multiplicative and implicative, are proposed and discussed. The purpose of this paper is to facilitate the choice and application of such operators through providing a systematization of their classes according to their behavior and equipping some classical averaging operators, namely the Power Means and the OWA operators, with importance weighting schemes and direct parametric andness control for both kinds of importance weighting. For increased efficacy and for symmetric behavior by andness and orness (= 1−andness) at the same degree of both measures, the two classes of averaging operators are applied in a De Morgan dual form. The main issue in the choice of underlying the classical averaging operator appears to be the computational cost of its application.