  A translation surface on (S, \Sigma) gives rise to two transverse measured
foliations \FF, \GG on S with singularities in \Sigma, and by integration, to a
pair of cohomology classes [\FF], \, [\GG] \in H^1(S, \Sigma; \R). Given a
measured foliation \FF, we characterize the set of cohomology classes \B for
which there is a measured foliation \GG as above with \B = [\GG]. This extends
previous results of Thurston and Sullivan.
  We apply this to two problems: unique ergodicity of interval exchanges and
flows on the moduli space of translation surfaces. For a fixed permutation
\sigma \in \mathcal{S}_d, the space \R^d_+ parametrizes the interval exchanges
on d intervals with permutation \sigma. We describe lines \ell in \R^d_+ such
that almost every point in \ell is uniquely ergodic. We also show that for
\sigma(i) = d+1-i, for almost every s>0, the interval exchange transformation
corresponding to \sigma and (s, s^2, \ldots, s^d) is uniquely ergodic.
  As another application we show that when k=|\Sigma| \geq 2, the operation of
`moving the singularities horizontally' is globally well-defined. We prove that
there is a well-defined action of the group B \ltimes \R^{k-1} on the set of
translation surfaces of type (S, \Sigma) without horizontal saddle connections.
Here B \subset \SL(2,\R) is the subgroup of upper triangular matrices.
