  In this paper, we investigate representations of links that are either
centrally symmetric in $\mathbb{R}^3$ or antipodally symmetric in
$\mathbb{S}^3$. By using the notions of antipodally self-dual and antipodally
symmetric maps, introduced and studied by the authors, we are able to present
sufficient combinatorial conditions for a link $L$ to admit such
representations. The latter naturally arises sufficient conditions for $L$ to
be amphichiral.
  We also introduce another (closely related) method yielding again to
sufficient conditions for $L$ to be amphichiral. We finally prove that a link
$L$, associated to a map $G$, is amphichiral if the self-dual pairing of $G$ is
not one of 6 specific ones among the classification of the 24 self-dual pairing
$Cor(G) \rhd Aut(G)$.
