Proving no simplex has 2-fold symmetry

[zooxanthellae]

Heck, I don't even know if this is true. Intuition seems to suggest that it is but I know of no way to prove it. In general I don't know of any way to rigorously determine axes of symmetry. And I am having trouble finding anything about this, so any links are appreciated.

 

[zooxanthellae]

Is there something unclear/asinine about my question? I fear it may be a little vague.

 

[fzero]

It is a bit vague. If you allow reflections through arbitrary points, then most simplices have reflection symmetry. Ex: reflection through the midpoint of a 1-simplex. If you have something else in mind, then you should state it more clearly.