Why is CO2 not in the C∞v point group? It seems to me that CO2, since it can be linearly rotated around the z-axis without change of shape, ought to be in the C∞v point group.  However, in all the character tables I can find, it's listed as a D∞h.  What's the difference between these two, and why does CO2 fall in the D∞h and not the C∞v?
Both are linear with a $C_{\infty}$ axis, but $D_{\infty h}$ has a center of inversion and $C_{\infty v}$ does not.