When wave vortices, that is nodal lines of a complex scalar wavefunction in space, approach transversely, their typical crossing and reconnection is a two-stage process incorporating two well-understood elementary events in which locally coplanar hyperbolas switch branches. The explicit description of this reconnection is a pedagogically useful illustration of several concepts: singularity, codimension, unfolding, genericity, bifurcation and normal forms. (Contains 5 figures.)