  An ordered and oriented 2-component link L in the 3-sphere is said to be
achiral if it is ambient isotopic to its mirror image ignoring the orientation
and ordering of the components. Kirk-Livingston showed that if L is achiral
then the linking number of L is not congruent to 2 modulo 4. In this paper we
study orientation-preserving or reversing symmetries of 2-component links,
spatial complete graphs on 5 vertices and spatial complete bipartite graphs on
3+3 vertices in detail, and determine the necessary conditions on linking
numbers and Simon invariants for such links and spatial graphs to be symmetric.
