  By studying the weak closure of multidimensional off-diagonal self-joinings
we provide a criterion for non-isomorphism of a flow with its inverse, hence
the non-reversibility of a flow. This is applied to special flows over rigid
automorphisms. In particular, we apply the criterion to special flows over
irrational rotations, providing a large class of non-reversible flows,
including some analytic reparametrizations of linear flows on the two torus, so
called von Neumann's flows and some special flows with piecewise polynomial
roof functions.. A topological counterpart is also developed with the full
solution of the problem of the topological self-similarity of continuous
special flows over irrational rotations. This yields examples of continuous
special flows over irrational rotations without topological self-similarities
and having all non-zero real numbers as scales of measure-theoretic
self-similarities.
