Circumstances leading to twisting crystallographic orientation in banded polymer spherulites are analyzed, both from the standpoint of qualitative morphology and of more quantitative measurement and calculation. Particular attention is paid to linear polyethylene, for which the most extensive information is available, and to alpha-poly(vinylidene fluoride), which behaves in a closely similar manner. In polymers that exhibit vigorous twisting (band spacings extending down to 10 mu m or less), chiral factors of two distinct kinds can be recognized. One is enantiomorphism, and the other is chain tilt (molecular stems in chain-folded crystals not being codirectional with lamellar normals). Quantitative evidence strongly suggests that the latter produces twisting orientation by generating surface forces in lamellae. It is also responsible for formation of dislocations that are isochiral, although these generally contribute relatively little to twisting. Banding in various classes of polymer is reviewed, and it is emphasized that there are several patterns of behavior, most still in need of elucidation; suggestions are made for continuing research.