XeF4 - why are there only 2 dihedral mirror planes?

I started studying molecular symmetry and I became somehow confused about *dihedral mirror planes*. I found a definition, which says, that it is a *vertical mirror plane, which bisects two $C\_2$ axes*.


But when I look at $\ce{XeF4}$ molecule, I see, that it has 4 $C\_2$ axes and 4 vertical mirror planes. All of those planes are bisecting 2 $C\_2$ axes and both all of them also go through some atom.


Still, there are only 2 dihedral mirror planes $\sigma\_\mathrm{d}$ and the remaining two planes are just common vertical ones $\sigma\_\mathrm{v}$.


So, could you please tell me what's the real difference between vertical and dihedral mirror planes?


OK, in a link given in [comments by Tyberius](https://chemistry.stackexchange.com/questions/87070/xef4-why-are-there-only-2-dihedral-mirror-planes#comment157285_87070) is said, that dihedral planes are such planes, which bisects as many bonds as possible, while "normal" vertical planes bisects as many atoms as possible. Then I don't get, why ethene (see the picture below) doesn't have one dihedral plane, as one of them clearly bisects its central bond and we can't symmetrically bisect more bonds in this molecule.




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### $\ce{XeF4}$


[![enter image description here](https://i.stack.imgur.com/BHsIm.png)](https://i.stack.imgur.com/BHsIm.png)




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### Ethene


[![enter image description here](https://i.stack.imgur.com/6H3j9.png)](https://i.stack.imgur.com/6H3j9.png)




This is a situation of coordinate systems. In the $D\_\mathrm{4h}$ point group, as you've noticed, there are 4 $C\_{2}$ rotation axes in the plane. The easiest way to set the coordinate system is to have one pair be coincident with the $x$ and $y$ axes and the other pair coincident with the functions $x=y$ and $x=-y$.


Herein lies the issue: which pair is which? I think most people would agree that a reasonable choice is that the primary set should be coincident with the Cartesian axes. Indeed, an example [character table](http://symmetry.jacobs-university.de/cgi-bin/group.cgi?group=604&option=4) for this group makes this choice as well (these are the $C\_{2}'$ elements). This then defines which pair of mirror planes are $\sigma\_\mathrm{d}$ and which pair are $\sigma\_\mathrm{v}$.


This covers $\ce{XeF4}$ only, not ethene, because ethene belongs to the $D\_\mathrm{2h}$ point group.


Here's the [character table](http://symmetry.jacobs-university.de/cgi-bin/group.cgi?group=602&option=4) for that group. There's no dihedral mirror plane because the mirror planes perpendicular to the plane of molecule are coincident with the rotation axes.


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So it seems that the use of $\sigma\_\mathrm{v}$ vs $\sigma\_\mathrm{d}$ is basically a matter of convention when both terms can be applied. Really, even $\sigma\_\mathrm{h}$ vs $\sigma\_\mathrm{v}$ is somewhat arbitrary if there isn't a single clear principal axis. 


By a certain [convention](http://www-e.openu.ac.il/symmetry/symmetry-tutorial.html) $\sigma\_\mathrm{d}$ are vertical mirror planes that bisect $2C\_2$ axes. When there are multiple mirror planes that fit this criteria, those that cut through the most atoms are referred to as $\sigma\_\mathrm{v}$, while the others are called $\sigma\_\mathrm{d}$.


This again is somewhat arbitrary and can be ambiguous. In particular, for $D\_\mathrm{2h}$ molecules like ethene any of the mirror planes could fit the description for $\sigma\_\mathrm{h}$ depending on which $C\_2$ you say is the principal. For this point group, you can unambiguously label the axes and planes using the [Cartesian axes](http://symmetry.jacobs-university.de/cgi-bin/group.cgi?group=602&option=4)


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