Because of twisting correlation of crystallites along the radial direction, polyethylene spherulites are known to develop concentric band pattern. The mechanism of branching and reorientation of lamellar crystallites in the banded spherulites has been examined experimentally by optical and atomic force microscopies associated with quenching and chemical etching. The microscopic observation suggests a branching instability of lamellar crystals at the growth front of the spherulite. We propose a mechanism of consecutive branching and twisting reorientation of branches regulated by the inherent torsional stress expected for the banded spherulites and the branching instability. We have experimentally examined the relationships among the growth rate, the maximum lateral width of crystals at the growth front, and the period of bands for three different molecular weight fractions of polyethylene. The predicted relationship among them holds for the fractions.