Twofold elastic symmetry ($N=13$)

A rotation through $\pi$ about the third coordinate axis preserves the strain components $e_{11},e_{22},e_{33},e_{12}$ and reverses $e_{13},e_{23}$. A symmetry-invariant elastic stiffness tensor cannot couple the two parity spaces. Its independent symmetric blocks therefore contain $4\cdot 5/2+2\cdot 3/2=13$ coefficients. A twofold axis alone does not impose invariance under every rotation about that axis.