  We give conditions for when continuous orbit equivalence of one-sided shift
spaces implies flow equivalence of the associated two-sided shift spaces. Using
groupoid techniques, we prove that this is always the case for shifts of finite
type. This generalises a result of Matsumoto and Matui from the irreducible to
the general case. We also prove that a pair of one-sided shift spaces of finite
type are continuously orbit equivalent if and only if their groupoids are
isomorphic, and that the corresponding two-sided shifts are flow equivalent if
and only if the groupoids are stably isomorphic.
  As applications we show that two finite directed graphs with no sinks and no
sources are move equivalent if and only if the corresponding graph
$C^*$-algebras are stably isomorphic by a diagonal-preserving isomorphism (if
and only if the corresponding Leavitt path algebras are stably isomorphic by a
diagonal-preserving isomorphism), and that two topological Markov chains are
flow equivalent if and only if there is a diagonal-preserving isomorphism
between the stabilisations of the corresponding Cuntz-Krieger algebras (the
latter generalises a result of Matsumoto and Matui about irreducible
topological Markov chains to a result about general topological Markov chains).
  We also show that for general shift spaces, strongly continuous orbit
equivalence implies two-sided conjugacy.
