Let $(X,\tau)$ be a topological space. Let us call $x,y\in X$ swappable if there is $f:X\to X$ continuous such that $f(x)=y$ and $f(y)=x$. This relation is obviously reflexive and symmetric, but not necessarily transitive.
Moreover, we call $(X,\tau)$ rigid if the identity is the only homeomorphism from $X$ to itself.
Is there a rigid space $(X,\tau)$ with $|X| > \aleph_0$ such that for all $x,y\in X$ we have that $x,y $ are swappable?
The answer is yes by a 1951 result of Miroslav Katětov, who proved that there is an (uncountable) rigid totally disconnected compact space. Any such space $X$ is rigid and has the property that any two points are swappable by continuous functions, since if $x\neq y$, we can find an open separation $X=U\sqcup V$ with $x\in U$ and $y\in V$, and then let $f$ be constant on each piece with value $y$ or $x$, respectively. This is a continuous function swapping $x$ and $y$.
