There's a distinction to be made between two notions: foliations and distributions.
A distribution is the data, at each point m of M, of a subspace of Tm(M). These subspaces are all of the same dimension (say r), and depend smoothly on the point m, which means that they are generated by r smooth vector fields.
A foliation is a partition of the manifold into (not necessarily closed) submanifolds, such that, locally, this partition looks like the standard decomposition of ℝn into translates of ℝd. Ok, there's a caveat in my description since a same leaf could come infinitely many often in the neighborhood of a given point m. Anyways... I'm assuming that you know what a foliation is.
Foliations of M form a subset of distributions on M.
The Frobenius integrability criterion (mentioned by Tom in him remark) states that a distribution D comes from a foliation iff for any vector fields v and w tangent to D, their Lie bracket is again tangent to D.
It turns out that that criterion is always satisfied for one-dimensional distributions, and so one-dimensional distributions are indeed in bijection with one-dimensional foliations. But that's no longer true for r ≥ 2.
The operation of taking orthogonal complement is a very good operation for distributions: it's always well defined, and the orthogonal complement of the orthogonal complement is the distribution you started with.
But the orthogonal complement of a foliation is typically only a distribution.
The standard example that illustrates that situation is the vector field sin(z)d/dx + cos(z)d/dy on ℝ3. It defines a perfectly good foliation, but its orthogonal fails to satisfy the Frobenius integrability criterion, and therefore fails to be a foliation (in this particular case, it's a contact structure, another beautiful mathematical notion...).
Ah! You also wanted the foliation to be defined by the orbits of a group acting by isometries... That can be arranged: take the action of S1 on S3 given by the Hopf fibration. The orthogonal distribution is the standard contact structure on S3.
You also said that you wanted you Riemanninan manifold to be flat... In that case, you can take ℝ4=ℂ2 with its S1-action by complex multiplication. That example contains the above S3 example as an
invariant submanifold, and therefore reproduces all its features.