  We say that a Euclidean lattice in $\mathbb R^n$ is permutation invariant if
its automorphism group has non-trivial intersection with the symmetric group
$S_n$, i.e., if the lattice is closed under the action of some non-identity
elements of $S_n$. Given a fixed element $\tau \in S_n$, we study properties of
the set of all lattices closed under the action of $\tau$: we call such
lattices $\tau$-invariant. These lattices naturally generalize cyclic lattices
introduced by Micciancio, which we studied in a recent paper. Continuing our
investigation, we discuss some basic properties of permutation invariant
lattices, in particular proving that the subset of well-rounded lattices in the
set of all $\tau$-invariant lattices in $\mathbb R^n$ has positive co-dimension
(and hence comprises zero proportion) for all $\tau$ different from an
$n$-cycle.
