It was trying to undertsand the quantity that would define chirality for sufficiently smooth surfaces in $\mathbb{R}^3$. A motivation for this is the "notion" of handedness defined for a ruled surface \begin{equation} \mathbf{x}(s,v) = \mathbf{d}(s)+v\mathbf{g}(s), \end{equation}
where $\mathbf{d}$ is the directrix and $\mathbf{g}$ is the generatrix or the ruling of the ruled surface.The handedness is usually defined by the sign of the determinant : \begin{equation} det[\mathbf{\dot{d}}, \mathbf{g}, \mathbf{\dot{g}}]. \end{equation}
But even here the determinant vanishes for a developable surface, although I intuitively feel that the notion of left handed or right handed would still be valid for developable surfaces which are not planar. I am not sure how one would quantify this.
In general, if I have a surface of revolution or other surfaces, how would chirality be defined?
In geometry in general, a subset of a Euclidean space is said to be chiral or have chirality if it is not congruent to its mirror image.
That is, if there does not exist a rotation and translation of the mirror object that would make it exactly align with the original.
For surfaces of revolution, this amounts to the object being symmetric front to back down the axis of revolution. Specifically, if we generate a surface S by revolving the graph of $f:[a,b]\rightarrow\mathbb{R} $ around the x axis, then S is chiral if and only if $f(b-(x-a))\neq f(x)$ for some $x\in[a,b]$