  Interval graphs and interval orders are deeply linked. In fact, edges of an
interval graphs represent the incomparability relation of an interval order,
and in general, of different interval orders. The question about the conditions
under which a given interval graph is associated to a unique interval order (up
to duality) arises naturally. Fishburn provided a characterisation for uniquely
orderable finite connected interval graphs. We show, by an entirely new proof,
that the same characterisation holds also for infinite connected interval
graphs. Using tools from reverse mathematics, we explain why the
characterisation cannot be lifted from the finite to the infinite by
compactness, as it often happens.
