Dragon curve (thing)

In Jurassic Park, at each chapter heading, there is a fractal. You would find at chapter one, for instance, the following:

                 _____                                      
                      |
                      |
     _____       _____|
    |     |     |
    |     |     |
    |_____|_____|
          |
          |
     _____|
    |
    |
    |_____      *
          |     |
          |     |
          |_____|

      Iteration 1

The next one was more complex, and so on, and so forth. The astute among you would notice that this pattern is fractal in shape. This is a series known as a dragon curve, or, as the title of the node formerly suggested, the Jurassic Park Fractal, thanks to Crichton's attention given to it. Now, every fractal has a pattern of some sort, so let's look at this one.

First, I seriously doubt that this was in fact the "first iteration" of this thing. It's way too complex for that. Now, we know that the length of each line segment is the same, so that's not helping us much. So let's look at the turns. It would appear that it goes, from the start(labeled with an asterisk above):

Right? Anyway, let's see if we have any sort of symmetry here. We have an odd number of turns, so that means there's a dividing one, which I have bolded above. To each side of it, it appears that we have the opposite on each, right for left, and so on and so forth. So that means that the first iteration is something like this:

         *
         |
         |
    _____|

Hmm... that's just Right, it acting as its own center. Now, I'm going to cheat a little bit, and give you a freebie, but it's something you'd find if you compared Crichton's first and second iterations. It's that each iteration starts with the pattern of turns of the one before it... so iteration two starts off Right, Right, then of course, Left, balancing the sentence.

These are all the rules of the aforementioned fractal... but I'll give them to you again for simplicity's sake:

1. The center turn is always Right
2. The first half of the order is always the full order of the last iteration.
3. The second half is the exact reverse of the first. (See Right,Right,Left)

There is indeed an alternate way of determining the second half which is far more mathematical and useful for programming, say. The second half, if you notice, is the first half repeated, but with one change, that of the 2^n-2th term of the original sequence, i.e.:
Right, Right, Left, Right, Right, Left, Right

Clearly, this is useless when n=1, but hey, when n=1 the fractal is just Right, setting yourself up for a classic recursive function.