I guess you are looking for a nice answer, but here is a stupid one. On the other hand I'm sure that there is no "nice answer".
Say assume your $n$-gons are convex hulls
A polygon can is uniquely determined by length of ${ z_1,z_2,\dots,z_n }sides $\ell_i$ and ${ w_1,w_2,\dots,w_n}$. You may think parametrize plane by angles $\mathbb C$. You can assume that \alpha_i$. Thus we have to find a complete set of invariants for sequence $\sum w_i=\sum z_i=0$(\alpha_1,\ell_1,\alpha_2,\ell_2,\dots\alpha_n,\ell_n)$ which survive after even cyclic shifts and reversing order.
Then if polygons are equal then $|\sum z_i^k|=|\sum w_i^k|$ you prepare symmetric polynomials for any $k\ge 1$your group. Moreover Say take all monomials of degree at most one in each $(\sum z_i^k)^\ell(\sum w_i^\ell)^k=(\sum w_i^k)^\ell(\sum z_i^\ell)^k$ for any \alpha_i$ and $k,\ell\ge 1$\ell_i$ and take its mean value it along the group. You can also think that obtain a big collection of polynomial expressions in $k,\ell\le n$\alpha_i$ and $\ell_i$ which gives complete invariant (perimeter will be one of them).
I bet that is enough ;)
