A permutation $\sigma\in\mathfrak S_n$ is graceful if $\{|\sigma(i+1)-\sigma(i)| \text{ with } 1\leq i\leq n\}=\{1,2,\ldots,n-1\}$ (terminology coming from a more general definition in graph theory).
From the definition, it follows that if $\sigma$ is graceful and we set $\tau(i)=|\sigma(i+1)-\sigma(i)|$ for $i=1,\ldots,n-1$, then $\tau$ is a permutation (in $\mathfrak S_{n-1}$). This made me wonder about those $\sigma$'s such that $\tau$ is again graceful – I would call them doubly graceful. For instance, this is clearly the case for $\sigma=[1,3,2]$. I ran tests showing there are no doubly graceful permutations for $n=4$ to 9.
I could not find anything in the literature and I'm no expert of permutations nor graphs, so...
Question 1: Are there any doubly graceful permutations for $n>3$? (I hope this is not too trivial)
Question 2: Has anybody considered this before?