Let $X$ be a "nice" algebraic curve over some field $K$, say characteristic zero.
The twists of $X$, i.e., the curves $Y$ over $K$ such that $X_{\overline K} \cong Y_{\overline K}$, are in bijection with the continuous cohomology pointed set $H^1(\mathrm{Gal}(\overline{K}/K),X).$
I was just wondering about an analogous question for "sections" of curves.
Now, let $x\in X(K)$. Define a twist of $x$ to be a point $y\in X(K)$ such that $x_{\overline K}$ is "isomorphic" to $y_{\overline K}$. (I don't know if it makes sense to talk about "isomorphic" sections. But what I mean is that there is an automorphism $\sigma$ of $\overline K$ such that $x_{\overline K}\circ \sigma = y_{\overline K}$.
Q1. Are twists of $x$ in bijection with some "cohomology set"?
Q2. Do there even exist any non-trivial twists of $x$? I get a feeling that $x_{\overline K}\circ \sigma$ always descends to $x$ forcing $x=y$.