Blind Moving-Average (MA) parameters estimation methods often resort to higher-order-statistics (HOS) in the form of high-order moments or cumulants in order to retrieve the phase of the generating system when no phase information, e.g., minimum-phase, is available. In this work, a new generic statistic is proposed - called the characteristic mean or charm in short - a generalization of the ordinary mean vector, which nonetheless carries a special form of HOS. The charm is parameterized by a parameters-vector called processing-point, which, when properly selected, conveniently controls the trade-off between the charm's HOS information content and the variance of its sample-estimate. A blind charm-based iterative algorithm is proposed, involving data-driven selection of the processing-point. The resulting algorithm - called CHARMA - is shown to significantly outperform ordinary HOS-based algorithms.