So I gave Nifty a copy of Matt's code and told him to go away and study it. Nifty was not happy.

        bool EXTRA_TRIG_FUNCTION=false;
       
   float rxy;
   float ratrack=pi2-pi2/folds;   //pi2 is 2*pi
   float rotadjust=-pi/folds  + FoldAdjust;     // I called folds splits in my original code...
// I called them splits because I didn't know
   float pintrack=pi/folds;      //what they were called.....
   float pi2splits=pi2/folds;
   float  i=0.0;    //why a float??  I was doing something with it... 
   
   float omega=atan(z.y,z.x);
   if (omega<0.0) {omega+=pi2;}    //so angle goes from 0 to 2pi instead of having negative part
   
   float rotate=0;   

   //I only wanted to do the kaleidoscope rotations on specific iterations
  // doing multiple polyfolds... you can do it every iteration or whatever- for cool effects
  // only do it on specific iterations- I was making kaleidoscopes out of 2d polyfolding

   if (RotateIter==n) {rotate=Rotate;}    //only do the rotation on a specific iteration

   
   while (omega>pi2folds && i<floor(folds+1.0)) {
         i+=1.0;
         
         if (EvenFolds) {rotate*=-1;}    //if you have an even number of folds, you can do 1/2 as
                                                                 //many rotations- it's sort of cool

            if (!EXTRA_TRIG_FUNCTION) {   //that is a NOT "!" 
               rotadjust+=ratrack;
               pintrack+=pi2folds;
               omega-=pi2folds;
            } else {     //with extra trig function, you don't need the above... just do the above instead
               omega-=pi2folds;
            }
         }

         if (EXTRA_TRIG_FUNCTION) {   // don't do this unless you absolutely love using transcendental functions
             z=length(z)*vec2(cos(omega),sin(omega));
         }

// if you don't have an even number of folds, you have to make an even number of rotations like the following:

      if (omega>pi2folds*.5 && !EvenFolds) {rotate*=-1;}

// pintrack is used to shift the fold center outwords- images follow

         vec2 z2=vec2(cos(pintrack),sin(pintrack));   

// splitrad is the distance from the center that the folds are shifted outwards
      
         z.xy-=z2*splitrad;
          rxy=length (z.xy);   
         
// rotadjust keeps every duplicate aligned correctly.        
          omega=atan(z.y,z.x)+rotadjust+rotate+FoldAdjust;
         z.xy=rxy * vec2(cos(omega),sin(omega));


   return z;

   

  K... I explain momentarily. 

He then came back with this poor excuse for a polyfold menger.  Let me have a look at your code young man!

Then I remembered I had chosen not to use my brain this week, so I told Nifty you will have to seek guidance from Uncle Matt.

I noticeed Nifty was confused about rotate , "n" and how the "while"  loop works.

Whut? Is it still continuous? Or you have sliced it like sushi

The best try is just using Polyfold_sym as it is.
On even fold count discontinue on one half plane (not very visible). Else perfectly continue.
I remember the trick was to check the parity of the integer obtained from atan(). Change sign of y accordingly.
And probably in case of even folds.... do abs(y) on the sector n°zero. Not tested might be wrong

To be tried