An example to illustrate the difference between an inversion operation and a twofold rotation 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.
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