  The classification of isoparametric hypersurfaces in spheres with four or six
different principal curvatures is still not complete. In this paper we develop
a structural approach that may be helpful for a classification. Instead of
working with the isoparametric hypersurface family in the sphere, we consider
the associated Lagrangian submanifold of the real Grassmannian of oriented
$2$-planes in $\mathbb{R}^{n+2}$. We obtain new geometric insights into
classical invariants and identities in terms of the geometry of the Lagrangian
submanifold.
