  The {\em topological symmetry group} of an embedding $\Gamma$ of an abstract
graph $\gamma$ in $S^3$ is the group of automorphisms of $\gamma$ which can be
realized by homeomorphisms of the pair $(S^3, \Gamma)$. These groups are
motivated by questions about the symmetries of molecules in space. In this
paper, we find all the groups which can be realized as topological symmetry
groups for each of the graphs in the Heawood family. This is an important
collection of spatial graphs, containing the only intrinsically knotted graphs
with 21 or fewer edges. As a consequence, we discover that the graphs in this
family are all intrinsically chiral.
