You want to take a looping GIF, rotate the sequence of its frames so that the animation starts from the middle, and ensure that the GIF still loops seamlessly as if it were unchanged. Essentially, you're asking how to shift the order of the frames, so the animation's starting point is different, without breaking the continuous loop.

Here's how you can achieve that with ffmpeg:

1. Extract the frames of the original GIF.
2. Reorder the frames so that the middle frame becomes the first, while keeping the loop continuous.
3. Rebuild the GIF with the newly ordered frames.