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