  In this paper we present a new family of non-compact properly embedded,
self-shrinking, asymptotically conical, positive mean curvature ends
$\Sigma^n\subseteq\mathbb{R}^{n+1}$ that are hypersurfaces of revolution with
circular boundaries. These hypersurface families interpolate between the plane
and half-cylinder in $\mathbb{R}^{n+1}$, and any rotationally symmetric
self-shrinking non-compact end belongs to our family. The proofs involve the
global analysis of a cubic-derivative quasi-linear ODE. We also prove the
following classification result: a given complete, embedded, self-shrinking
hypersurface of revolution $\Sigma^n$ is either a hyperplane $\mathbb{R}^{n}$,
the round cylinder $\mathbb{R}\times S^{n-1}$ of radius $\sqrt{2(n-1)}$, the
round sphere $S^n$ of radius $\sqrt{2n}$, or is diffeomorphic to an $S^1\times
S^{n-1}$ (i.e. a "doughnut" as in [Ang], which when $n=2$ is a torus). In
particular for self-shrinkers there is no direct analogue of the Delaunay
unduloid family. The proof of the classification uses translation and rotation
of pieces, replacing the method of moving planes in the absence of isometries.
