If L is a relational language, then an L-structure \({\mathbb {X}}\) is reversible iff each bijective homomorphism (condensation) \(f:{\mathbb {X}}\rightarrow {\mathbb {X}}\) is an automorphism of \({\mathbb {X}}\) . We show that \({\mathbb {X}}\) is not reversible iff there is a back and forth system \(\Pi \) of partial self-condensations of \({\mathbb {X}}\) containing one which is not a partial isomorphism and having certain closure properties. Using that characterization we detect several classes of non-reversible partial orders containing, for example, homogeneous-universal posets (in particular, the random poset), the divisibility lattice, \(\langle {\mathbb {N}},\,\mid \,\rangle \) , the ideals \([\kappa ]^{<\lambda }\) , the meager ideal in the algebra \({\textrm{Borel}}(\omega ^\omega )\) , and the direct powers of rationals, \({\mathbb {Q}}^\kappa \) , and integers, \({\mathbb {Z}}^\kappa \) . Some of the results are obtained under additional set-theoretic assumptions.
