Q:

Difference between 運命 and 命運

To my knowledge, 運命{うんめい} is the more common of the two terms, meaning fate / destiny.  命運{めいうん} is clearly the same 漢字 reversed, and translations give the same meaning of fate / destiny.  
I was wondering if the two are in fact the exact same in meaning, or if there's a difference in nuance between them, other than the fact that 運命{うんめい} seems to be the more common of the two. 

A:

大辞林 says

命運
  そのこと（もの）の存続にかかわる重大な運命。「—が尽きる」「国家の—」

the keywords being 存続にかかわる重大な, that is 命運 is serious and may affect the continuance of the thing or person whose 命運 is being discussed.  WWWJDICT gives "doom" as translation; I don't think it is a good translation, but thematically it fits very well, it's a kind of fate that may be the last fate the thing or person ever sees. 
As for the word itself, I wouldn't see 命運 as a switched version of 運命, rather as something like

命運 ≒ 命【いのち】にかかわる運【うん】 (or 命にかかわる運命)

(運 is a word by itself), i.e. the life-or-death kind of fate. 
Lastly, just to quantify your "more common", the BCCWJ has

運命 3260 results
  命運 236 results

