Test a polyomino against Conway criterion
Python 3.8 (pre-release), 371 ... 338 336 bytes

Takes as input a list of complex numbers, denoting the boundary coordinates in counterclockwise order.

-9 bytes thanks to @ovs
-2 bytes thanks to @Bubbler



import itertools as Z
def f(P):Q=P*2;I=Q.index;L=len;return any(L({*map(complex.__sub__,T:=(J:=lambda x,y:Q[(j:=I(p[x])):I(p[y],j)+1])(0,1),U:=J(3,4)[::-1])})&lt;2&lt;L({*p})and(L(T)==L(U))&amp;all((B:=J(a,-~a%6))==[B[L(B)//2]-E+B[~L(B)//2]for E in B][::-1]for a in[1,2,4,5])for S in zip(*[Q[I(i):]for i in P])for p in eval('Z.'+dir(Z)[11])(S,6))


Try it online!