Extending the Symmetry Group: Even/Odd Z2 Symmetry?

[amrashed]

I am extending the standard model symmetry group by introducing discrete symmetry (Z2). The group could be (I, G) or (I, -G). Is that called even and odd Z2 symmetry? What is the difference of considering either of them?

 

[chrispb]

There is exactly one finite group of order 2.