  In four dimensional N=2 supergravity theories, BPS bound states near marginal
stability are described by configurations of widely separated constituents with
nearly parallel central charges. When the vacuum moduli can be dialed
adiabatically until the central charges become anti -parallel, a paradox
arises. We show that this paradox is always resolved by the existence of "bound
state transformation walls" across which the nature of the bound state changes,
although the index does not jump. We find that there are two distinct phenomena
that can take place on these walls, which we call recombination and
conjugation. The latter is associated to the presence of singularities at
finite distance in moduli space. Consistency of conjugation and wall-crossing
rules near these singularities leads to new constraints on the BPS spectrum.
Singular loci supporting massless vector bosons are particularly subtle in this
respect. We argue that the spectrum at such loci necessarily contains massless
magnetic monopoles, and that bound states around them transform by intricate
hybrids of conjugation and recombination.
