Ramseyan variations on symmetric subsequences

A theorem of Dekking in the combinatorics of words implies that there exists an injective order-preserving transformation $f:\{0,1,\dots,n\}\to\{0,1,\dots,2n\}$ with the restriction $f(i+1)\le f(i)+2$ such that for every 5-term arithmetic progression $P$ its image $f(P)$ is not an arithmetic progression. In this paper we consider symmetric sets in place of arithmetic progressions and prove lower and upper bounds for the maximum $M=M(n)$ such that every $f$ as above preserves the symmetry of at least one symmetric set $S\subseteq\{0,1,\dots,n\}$ with $|S|\ge M$.