The problem of aggregating multiple criteria to form an overall measure is of considerable importance in many disciplines. The ordered weighted averaging (OWA) aggregation, introduced by Yager, uses weights assigned to the ordered values rather than to the specific criteria. This allows one to model various aggregated preferences, preserving simultaneously the impartiality (neutrality) with respect to the individual criteria. However, importance weighted averaging is a central task in multicriteria decision problems of many kinds. It can be achieved with the Weighted OWA (WOWA) aggregation, introduced by Torra, covering both the weighted means and the OWA averages as special cases. In this paper we analyze the monotonicity properties of the WOWA aggregation with respect to changes of importance weights. In particular, we demonstrate that a rank reversal phenomenon may occur in the sense that increasing the importance weight for a given criterion may enforce the opposite WOWA ranking than that imposed by the criterion values.