We propose a new interpretation of the relationships between the dynamo cycle period (Pcyc) as observed in Ca II H and K, the rotational period (Prot), the activity level, and other stellar properties. Viewed within this framework, the data suggest that the dynamo α-parameter increases with magnetic field strength, contrary to the conventional idea of α-quenching. The data also suggest a possibly discontinuous dependence of the ratio of cycle to rotation frequency, ωcyc/Ω, as a function of Rossby number, Ro (or equivalently, activity or age). Stars evolve with ωcyc/Ω~t-0.35 (or Ro-0.7), until age t~2-3 Gyr (roughly at the Vaughan-Preston gap), where a sharp transition occurs, in which ωcyc/Ω increases by a factor of ~6. Thereafter, evolution with ωcyc/Ω~t-0.35 continues. The age at which transition occurs may be mass dependent, with K stars making the transition first.