2-bridge links $L(p/q)$ are described by the continuous fraction expansion $\frac{p}{q}=\left[a_1,a_2,\ldots,a_n\right]$, where the $a_i$ are the numbers of twists in the boxes below:

Looking at some knot table I've got the impression that a 2-bridge knot has vanishing Chern-Simons invariant if and only if the continued fraction expansion is symmetric in the sense that $$a_1=a_n,a_2=a_{n-1},\ldots.$$ For example $$5/2=\left[2,2\right], 13/5=\left[2,1,1,2\right], 17/4=\left[4,4\right], 25/7=\left[3,1,1,3\right]$$ $$29/12=\left[2,2,2,2\right], 41/9=\left[4,1,1,4\right], \ldots, 149/44=\left[3,2,1,1,2,3\right],\ldots$$ all have vanishing Chern-Simons invariant.

Question: is this true, is it known and what is a citeable reference?

(I would already be happy with the "if"-part, i.e., that all these knots satisfy $CS=0$.)