  We give a necessary and sufficient condition for a homogeneous Markov process
taking values in $\R^n$ to enjoy the time-inversion property of degree
$\alpha$. The condition sets the shape for the semigroup densities of the
process and allows to further extend the class of known processes satisfying
the time-inversion property. As an application we recover the result of
Watanabe in \cite{Wa1975} for continuous and conservative Markov processes on
$\R_+$. As new examples we generalize Dunkl processes and construct a
matrix-valued process with jumps related to the Wishart process by a
skew-product representation.
