  We consider the family of guarded and unguarded ordered logics, that
constitute a recently rediscovered family of decidable fragments of first-order
logic (FO), in which the order of quantification of variables coincides with
the order in which those variables appear as arguments of predicates.
  While the complexities of their satisfiability problems are now
well-established, their model theory, however, is poorly understood.
  Our paper aims to provide some insight into it.
  We start by providing suitable notions of bisimulation for ordered logics.
  We next employ bisimulations to compare the relative expressive power of
ordered logics, and to characterise our logics as bisimulation-invariant
fragments of FO a la van Benthem.
  Afterwards, we study the Craig Interpolation Property~(CIP).
  We refute yet another claim from the infamous work by Purdy, by showing that
the fluted and forward fragments do not enjoy CIP.
  We complement this result by showing that the ordered fragment and the
guarded ordered logics enjoy CIP.
  These positive results rely on novel and quite intricate model constructions,
which take full advantage of the "forwardness" of our logics.
