  Linear spin wave theory (LSWT) is the standard technique to compute the
spectra of magnetic excitations in quantum materials. In this paper, we show
that LSWT, even under ordinary circumstances, may fail to implement the
symmetries of the underlying ordered magnetic Hamiltonian leading to spurious
degeneracies. In common with pseudo-Goldstone modes in cases of quantum
order-by-disorder these degeneracies tend to be lifted by magnon-magnon
interactions. We show how, instead, the correct symmetries may be restored at
the level of LSWT. In the process we give examples, supported by
nonperturbative matrix product based time evolution calculations, where
symmetries dictate that there should be a topological magnon gap but where LSWT
fails to open up this gap. We also comment on possible spin split magnons in
MnF$_2$ and similar rutiles by analogy to recently proposed altermagnets.
