An exercise of Ziegler's book, Lectures on Polytopes, says:
Let $x(t)=(t,t^2,\dots, t^d) $ be the $d $ dimensional moment curve.Consider the cyclic polytope $C_d(n)=conv\{x(0),\dots,x(n-1)\} $. Show that there is an affine symmetry (an affine reflection) which induces the symmetry $x(i-1)\longleftrightarrow x(n-i) $, and thus the corresponding combinatorial symmetry of the cyclic polytope.
Here what does the induced symmetry means? It seems quite obvious that any reflection through a hyperplane does not give a translated, rotated, scaled copy of the moment curve passing through the image of the vertices. If my guess is wrong, how can I construct such a hyperplane and curve? If it is right, then what does the problem want me to do?