  Order-invariant first-order logic is an extension of first-order logic (FO)
where formulae can make use of a linear order on the structures, under the
proviso that they are order-invariant, i.e. that their truth value is the same
for all linear orders. We continue the study of the two-variable fragment of
order-invariant first-order logic initiated by Zeume and Harwath, and study its
complexity and expressive power. We first establish coNExpTime-completeness for
the problem of deciding if a given two-variable formula is order-invariant,
which tightens and significantly simplifies the coN2ExpTime proof by Zeume and
Harwath. Second, we address the question of whether every property expressible
in order-invariant two-variable logic is also expressible in first-order logic
without the use of a linear order. While we were not able to provide a
satisfactory answer to the question, we suspect that the answer is ``no''. To
justify our claim, we present a class of finite tree-like structures (of
unbounded degree) in which a relaxed variant of order-invariant two-variable FO
expresses properties that are not definable in plain FO. On the other hand, we
show that if one restricts their attention to classes of structures of bounded
degree, then the expressive power of order-invariant two-variable FO is
contained within FO.
