  We show that if a nontrivial group admits a locally invariant ordering, then
it admits uncountably many locally invariant orderings. For the case of a
left-orderable group, we provide an explicit construction of uncountable
families of locally invariant orderings; for a general group we provide an
existence theorem that applies compactness to yield uncountably many locally
invariant orderings. Along the way, we define and investigate the space of
locally invariant orderings of a group, the natural group actions on this
space, and their relationship to the space of left-orderings.
