The structure of pathological instances for simplex algorithms
Amenta and Ziegler proved that all currently-known constructions of exponential-time instances for simplex follow a particular structure that they call "deformed products":


  Deformed Products and Maximal Shadows of Polytopes by Amenta and Ziegler


However, I don't think there's any reason to believe that all bad instances for simplex have this structure.  This is probably just an artifact of the research process:


Klee and Minty found the first exponential-time example.
Other researchers looked and Klee and Minty's techniques and extended them to other pivot rules.  They naturally took the path of least resistance and followed the Klee-Minty cube as closely as possible.
Once someone finds one bad example for a pivot rule, there's no incentive for people to look for more.  As a result, all the bad examples we know of have a similar structure.