YES

Problem:
 f(g(x)) -> g(g(f(x)))
 f(g(x)) -> g(g(g(x)))

Proof:
 Arctic Interpretation Processor:
  dimension: 2
  interpretation:
             [1  0 ]  
   [f](x0) = [-& 1 ]x0,
   
             [0 0]  
   [g](x0) = [0 0]x0
  orientation:
             [1 1]     [1 1]              
   f(g(x)) = [1 1]x >= [1 1]x = g(g(f(x)))
   
             [1 1]     [0 0]              
   f(g(x)) = [1 1]x >= [0 0]x = g(g(g(x)))
  problem:
   f(g(x)) -> g(g(f(x)))
  String Reversal Processor:
   g(f(x)) -> f(g(g(x)))
   Matrix Interpretation Processor: dim=4
    
    interpretation:
               [1 0 0 1]     [0]
               [0 1 1 0]     [0]
     [f](x0) = [0 0 1 1]x0 + [1]
               [0 0 1 1]     [0],
     
               [1 1 1 0]  
               [0 0 0 1]  
     [g](x0) = [0 0 1 0]x0
               [0 0 0 1]  
    orientation:
               [1 1 2 2]    [1]    [1 1 2 2]    [0]             
               [0 0 1 1]    [0]    [0 0 1 1]    [0]             
     g(f(x)) = [0 0 1 1]x + [1] >= [0 0 1 1]x + [1] = f(g(g(x)))
               [0 0 1 1]    [0]    [0 0 1 1]    [0]             
    problem:
     
    Qed