Ring periodic structures are widely spread in engineering applications, where natural frequencies can split due to the deviation from axisymmetry. This work aims at the elimination of the dual-mode natural frequency splitting by using feature shifting. A concept of equivalent feature is introduced based on the grouping idea and an analytical model with discrete features is established. The relationships between the group number, feature number, shifting angle and excited circumferential wavenumber are identified as closed-form expressions. The splitting for structures with unshifted standard features depends on the feature number and wavenumber regardless of the grouping patterns. The equivalence between various groupings is verified. Simple rules governing the dual-mode splitting are elaborated, where one splitting is removed by a combination of equivalent feature number and wavenumber, and the other is eliminated by feature shifting. The rules allow immediate estimation and elimination of the dual-mode splitting where the modified structures still hold symmetry. The results can find application in vibratory structures where the frequency splitting is the key concern. The main results are verified by the Finite Element method.