  Let $X$ be a regular linear diffusion whose state space is an open interval
$E\subseteq\mathbb{R}$. We consider a diffusion $X^*$ which probability law is
obtained as a Doob $h$-transform of the law of $X$, where $h$ is a positive
harmonic function for the infinitesimal generator of $X$ on $E$. This is the
dual of $X$ with respect to $h(x)m(dx)$ where $m(dx)$ is the speed measure of
$X$. Examples include the case where $X^*$ is $X$ conditioned to stay above
some fixed level. We provide a construction of $X^*$ as a deterministic
inversion of $X$, time changed with some random clock. The study involves the
construction of some inversions which generalize the Euclidean inversions.
Brownian motion with drift and Bessel processes are considered in details.
