As discussed in the relevant Wikipedia article, beta decay via electron capture is possible in circumstances when positron emission is not.
From a simple (and perhaps naive) point of view, the two decay modes should be $$ ^A_ZX + e^- \rightarrow \ ^A_{Z-1}Y+ \nu_e $$ (for electron capture) and $$ ^A_ZX \rightarrow \ ^A_{Z-1}Y+e^+ + \nu_e $$ (in the case of positron emission).
Clearly, these two reactions are related through crossing. However, there are isotopes in nature which permit the first but not the second. This would appear to violate crossing symmetry since an allowed reaction is related to a forbidden reaction through crossing.
The usual heuristic explanation for why positron emission is sometimes energetically forbidden is that the reduction in nuclear charge necessarily requires releasing one of the bound-state electrons into the continuum (meaning that positron emission is really the emission of two particles not one). In that case, the reactions would look like $$ [^A_ZX]^{0+} \rightarrow \ [^A_{Z-1}Y]^{0+} + \nu_e $$ (in the case of electron capture) and $$ [^A_ZX]^{0+} \rightarrow \ [^A_{Z-1}Y]^{0+} + e^- + e^+ + \nu_e $$ (for positron emission), which are obviously not related through crossing.
But this way of viewing the reaction raises several questions: Would a bare nucleus be able to decay through positron emission more easily? Does the weak force still mediate these atomic decays which seem to actively involve the "spectator" electrons?
More seriously, what is preventing a "two step" reaction that begins with the emission of a lone positron but is followed by a separate "relaxation" reaction to release the electron? Alternatively, if the atom were surrounded by positive ions with which the negatively charged decay ion could easily form an ionic molecule, would electron-less positron emission become an allowed decay route?
There are many other related questions, but they all boil down to the same main issue:
Can the asymmetry between positron emission and electron capture be accounted for without violating crossing symmetry? And, if so, how?