  To a Toeplitz flow $(X,T)$ we associate an ordered $K^0$-group, denoted
$K^0(X,T)$, which is order isomorphic to the $K^0$-group of the associated
(non-commutative) $C^\ast$-crossed product $C(X)\rtimes_T \mathbb{Z}$. However,
$K^0(X,T)$ can be defined in purely dynamical terms, and it turns out to be a
complete invariant for (strong) orbit equivalence. We characterize the
$K^0$-groups that arise from Toeplitz flows $(X,T)$ as exactly those simple
dimension groups $(G,G^+)$ that contain a noncyclic subgroup $H$ of rank one
that intersects $G^+$ nontrivially. Furthermore, the Bratteli diagram
realization of $(G,G^+)$ can be chosen to have the ERS-property, i.e. the
incidence matrices of the Bratteli diagram have equal row sums. We also prove
that for any Choquet simplex $K$ there exists an uncountable family of pairwise
non-orbit equivalent Toeplitz flows $(X,T)$ such that the set of $T$-invariant
probability measures $M(X,T)$ is affinely homeomorphic to $K$, where the
entropy $h(T)$ may be prescribed beforehand. Furthermore, the analogous result
is true if we substitute strong orbit equivalence for orbit equivalence, but in
that case we can actually prescibe both the entropy and the maximal
equicontinuous factor of $(X,T)$. Finally, we present some interesting concrete
examples of dimension groups associated to Toeplitz flows.
