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