  Given an isometric immersion $f\colon M^n\to \R^{n+1}$ of a compact
Riemannian manifold of dimension $n\geq 3$ into Euclidean space of dimension
$n+1$, we prove that the identity component $Iso^0(M^n)$ of the isometry group
$Iso(M^n)$ of $M^n$ admits an orthogonal representation $\Phi\colon
Iso^0(M^n)\to SO(n+1)$ such that $f\circ g=\Phi(g)\circ f$ for every $g\in
Iso^0(M^n)$. If $G$ is a closed connected subgroup of $Iso(M^n)$ acting locally
polarly on $M^n$, we prove that $\Phi(G)$ acts polarly on $\R^{n+1}$, and we
obtain that $f(M^n)$ is given as $\Phi(G)(L)$, where $L$ is a hypersurface of a
section which is invariant under the Weyl group of the $\Phi(G)$-action. We
also find several sufficient conditions for such an $f$ to be a rotation
hypersurface. Finally, we show that compact Euclidean rotation hypersurfaces of
dimension $n\geq 3$ are characterized by their underlying warped product
structure.
