In the limit Hilbert curve, what percentage of order 1 Hilbert curves that appear in it are "right side up"?

HilbertCalc.cs

using System;

// In the limit Hilbert curve, what percentage of order 1 Hilbert curves that appear in it are "right side up"?
namespace HilbertCalc {
    class Program {
        static void Main(string[] args) {
            ulong[] curves = new ulong[] { 1, 0, 0, 0 }; // index 0 is "right side up," each subsequent index is 90 degrees counterclockwise
            for (int i = 1; i <= 20; i++) {
                Console.WriteLine($"Order {i} Hilbert curve:");
                ulong sum = curves[0] + curves[1] + curves[2] + curves[3];
                Console.WriteLine($"{curves[0]}/{sum} right side up");
                Console.WriteLine($"{curves[1]}/{sum} CCW");
                Console.WriteLine($"{curves[3]}/{sum} CW");
                Console.WriteLine($"{curves[2]}/{sum} upside down");
                // Each iteration of the curve contains:
                // two right-side-up copies of the previous iteration
                // one copy rotated clockwise
                // one copy rotated counterclockwise
                curves = new ulong[] {
                    curves[3] + 2 * curves[0] + curves[1],
                    curves[0] + 2 * curves[1] + curves[2],
                    curves[1] + 2 * curves[2] + curves[3],
                    curves[2] + 2 * curves[3] + curves[0],
                };
            }
            Console.ReadLine();
        }
        // It seems that the answer for all orientations is 25%. It's kind of like a modular "blur" convolution: each bucket ends up
        // iteratively bleeding into adjacent buckets, around and around in a circle, until all buckets are pretty much equally full.
        // In the limit, I imagine that all buckets ARE equally full.
    }
}