A detailed treatment is presented of the crystallography of twinning shears which cause the polymer chain axes in polymer crystals to be invariant or to be rotated but undistorted. These chain-axis invariant and rotation twins in polyethylene are considered in detail. The polyethylene twinning mode with twinning elements $$\begin{gathered} K_1 = \{ 102\} ;\eta _1 = \left\langle {20\bar 1} \right\rangle ; K_2 = \{ 100\} ; \hfill \\ \eta _2 = \left\langle {001} \right\rangle and magnitude of shear{\mathbf{ }}g = 0.34 \hfill \\ \end{gathered} $$ is proposed as the twinning mode which is most likely to cause a marked chain axis rotation without causing a change in the orientation of theb axis and only a small change in thickness of crystalline lamellae. These are the requirements ofc-twinning which was used byPietralla (Colloid & Polymer Sci.254, 249–257 (1976) to explain some aspects of the orientation behaviour of polyethylene.