  First we construct minimal hypersurfaces $M\subset\mathbf{R}^{n+1}$ in a
neighborhood of the origin, with an isolated singularity but cylindrical
tangent cone $C\times \mathbf{R}$, for any strictly minimizing strictly stable
cone $C$ in $\mathbf{R}^n$. We show that many of these hypersurfaces are area
minimizing. Next, we prove a strong unique continuation result for minimal
hypersurfaces $V$ with such a cylindrical tangent cone, stating that if the
blowups of $V$ centered at the origin approach $C\times \mathbf{R}$ at infinite
order, then $V = C\times\mathbf{R}$ in a neighborhood of the origin. Using this
we show that for quadratic cones $C = C(S^p \times S^q)$, in dimensions $n >
8$, all $O(p+1) \times O(q+1)$-invariant minimal hypersurfaces with tangent
cone $C\times \mathbf{R}$ at the origin are graphs over one of the surfaces
that we constructed. In particular such an invariant minimal hypersurface is
either equal to $C\times \mathbf{R}$ or has an isolated singularity at the
origin.
