Over-then-Under (OU) tangles are oriented tangles whose strands travel
through all of their over crossings before any under crossings. In this
paper we discuss the idea of {\em gliding\me}: an algorithm by which
tangle diagrams could be brought to OU form. By analyzing cases in which the algorithm converges, we obtain
a braid classification result, which we also extend to virtual braids,
and provide a Mathematica implementation. We discuss other instances
of successful ``gliding ideas'' in the literature --
sometimes in disguise -- such as the Drinfel'd double construction,
Enriquez's work on quantization of Lie bialgebras, and Audoux and
Meilhan's classification of welded homotopy links.