We derive the general anomaly polynomial for a class of two-dimensional CFTs arising as twisted compactifications of a higher-dimensional theory on compact manifolds, including the contribution of its isometries. We then use the result to perform a counting of microstates for dyonic rotating supersymmetric black strings in $\operatorname{AdS}_5 \times S^5$ and $\operatorname{AdS}_7 \times S^4$. We explicitly construct these solutions by uplifting a class of four-dimensional rotating black holes. We provide a microscopic explanation of the entropy of such black holes by using a charged version of the Cardy formula.