I cannot resist but mention a related concept, which in a sense generalizes the example you quote. 

Let a Lie group $G$ act properly and isometrically 
on the complete Riemannian manifold $M$. The action is called
**polar** if there exists a complete connected submanifold $\Sigma$ that meets all the orbits,
and meets them always orthogonally. Such a submanifold $\Sigma$ is called a **section**.
It is easy to see that a section must be totally geodesic. If an action admits a section
which is flat in the induced metric, then this action is called **hyperpolar**. 
In the case of linear orthogonal actions (or representations), there is no distinction between polar and hyperpolar actions since the complete totally geodesic submanifolds of Euclidean space are its affine subspaces.   

One example of polar representation which is very familiar from basic courses in linear algebra is the $SO(n)$-conjugation of $n\times n$  real symmetric matrices. It is well 
known that every symmetric matrix is orthogonally conjugate to a diagonal matrix, so here 
the section is given by the subspace of diagonal matrices. More generally,   
the standard examples of polar representations are the isotropy representations of symmetric spaces. Conversely, Dadok has shown that these essentially exhaust all the examples. 

The orbital foliation of a polar action has many remarkable geometric and topological properties. The story starts with Bott and Samelson in the 1950's and goes on. To mention
a recent result, A. Lytchak and G. Thorbergsson have proven that the orbifold points of the orbit space of a proper and isometric action correspond precisely to the points of the manifold where the slice representation is polar (J. Differential Geom.85 (2010), 117-140). 
Polar actions have also been generalized in many directions, e.g. polar foliations, 
complex actions, Hilbert space. I leave you with two book references: http://vmm.math.uci.edu/CriticalPointTheory.pdf (link to free book) and J. Berndt, S. Console and C. Olmos, Submanifolds and Holonomy, CRC/Chapman and Hall Research Notes Series in Mathematics 434 (2003), Boca Raton (more recent book).

**Edit:**Ah, I forgot to mention one detail. In your example, you removed the origin to get a second foliation by radial lines. For a polar action, the orbits form a (singular) foliation of course, but the sections form a foliation only if you remove the singular orbits. In fact, a point in the manifold lies in an orbit of lower dimension precisely if it is contained in more than one section.