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Reading in Atkin's Inorganic Chemistry book, I cam accross:

Although an inversion and a twofold rotation may sometimes achieve the same effect, that is not the case in general and the two operations must be distinguished.

He does not give an example of an occasion where such cases are not the same.

I will appreciate if someone can give an example of when inversion and twofold operation won't give the same result.

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If the body is chiral, a ${C_2}$-axis maintains chirality. An inversion (${S_2}$, ${i}$, the product of ${C_2}$ and ${σ}$h) inverts chirality.

https://ww2.chemistry.gatech.edu/~wilkinson/Class_notes/CHEM_3111_6170/Symmetry_and_group_theory.pdf
Frame 5
http://alpha.chem.umb.edu/chemistry/ch612/documents/MolecularGeometryandPointGroups_002.pdf
More technical
http://www.b-u.ac.in/sde_book/msc_phychem.pdf
The math

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  • $\begingroup$ Of course, chiral objects do not have any Sn improper axes of symmetry. Above describes operation of the transformation . $\endgroup$ – Uncle Al May 6 '14 at 15:47

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