ETA: Ah, yes, I think this was it, in outline: For a figure-of-eight symmetry, the orbit has to be asymptotically straight at (or near) the point at which it self-intersects. This requires the potential field at (or near) the point of self-intersection to be locally flat. Therefore, examine the nature of the three elementary cases that yield this: If the point is a local minimum, it means the orbit passes through or very near an attractor, which is unstable (small perturbation -> large effect). If the point is a local maximum, it means the orbit will tend to "fall off" it, which makes it unstable (ditto). If the point is a saddle, it is neutrally stable itself, but the relevant approaches to the point are necessarily "across slopes" which the orbit will want to "slide down" in the long run.