FIG. 1 is a back view showing my new design of a swiveling chain clip; FIG. 2 is a first side view showing my new design of a swiveling chain clip; FIG. 3 is a front view of my new design of a swiveling chain clip; FIG. 4 is a second side view of my new design of a swiveling chain clip taken on a side opposite the side shown in FIG. 2; FIG. 5 is a bottom view of my new design of a swiveling chain clip; and, FIG. 6 is a top view of my new design of a swiveling chain clip.