The spatial symmetry property of truncated birth-death 
processes studied in Di~Cre\-scenzo \cite{Di98} is extended 
to a wider family of continuous-time Markov chains. 
We show that it yields simple expressions for first-passage-time 
densities and avoiding transition probabilities, and apply it 
to a bilateral birth-death process with jumps. 
It is finally proved that this symmetry property is preserved 
within the family of strongly similar Markov chains.