  In this paper we study the action of a countable group $\Gamma$ on the space
of orders on the group. In particular, we are concerned with the invariant
probability measures on this space, known as invariant random orders. We show
that for any countable group the space of random invariant orders is rich
enough to contain an isomorphic copy of any free ergodic action, and
characterize the non-free actions realizable. We prove a Glasner-Weiss
dichotomy regarding the simplex of invariant random orders. We also show that
the invariant partial order on $\mathrm{SL}_3(\mathbf{Z})$ corresponding to the
semigroup of positive matrices cannot be extended to an invariant random total
order. We thus provide the first example for a partial order (deterministic or
random) that cannot be randomly extended.
