In $\mathbb R^3$ , the intersection of a plane and a sphere (e.g. $x^2 + y^2 + z^2 = 1$) is either empty, a single point, or a circle. All isometries of those circles are realized by isometries of the full sphere. In contrast, every plane intersects a cone (e.g. $x^2 + y^2 - z^2 = 0$) in a conic section which has reflection symmetries along one, two, or more axes. The one reflection is realized by an isometry of the cone, but the second, in general, is not.
What surfaces in $\mathbb R^3$ , or general subsets of $\mathbb R^3$ , are such that all non-empty planar intersections are either 1 point or have nontrivial symmetry? When are those symmetries not extensible to a symmetry of the full surface?