  The $^{150}$Nd($^3$He,$t$) reaction at 140 MeV/u and $^{150}$Sm($t$,$^3$He)
reaction at 115 MeV/u were measured, populating excited states in $^{150}$Pm.
The transitions studied populate intermediate states of importance for the
(neutrinoless) $\beta\beta$ decay of $^{150}$Nd to $^{150}$Sm. Monopole and
dipole contributions to the measured excitation-energy spectra were extracted
by using multipole decomposition analyses. The experimental results were
compared with theoretical calculations obtained within the framework of
Quasiparticle Random-Phase Approximation (QRPA), which is one of the main
methods employed for estimating the half-life of the neutrinoless $\beta\beta$
decay ($0\nu\beta\beta$) of $^{150}$Nd. The present results thus provide useful
information on the neutrino responses for evaluating the $0\nu\beta\beta$ and
$2\nu\beta\beta$ matrix elements. The $2\nu\beta\beta$ matrix element
calculated from the Gamow-Teller transitions through the lowest $1^{+}$ state
in the intermediate nucleus is maximally about half of that deduced from the
half-life measured in $2\nu\beta\beta$ direct counting experiments and at least
several transitions through $1^{+}$ intermediate states in $^{150}$Pm are
required to explain the $2\nu\beta\beta$ half-life.
  Because Gamow-Teller transitions in the $^{150}$Sm($t$,$^3$He) experiment are
strongly Pauli-blocked, the extraction of Gamow-Teller strengths was
complicated by the excitation of the $2\hbar\omega$, $\Delta L=0$, $\Delta S=1$
isovector spin-flip giant monopole resonance (IVSGMR). However, the near
absence of Gamow-Teller transition strength made it possible to cleanly
identify this resonance, and the strength observed is consistent with the full
exhaustion of the non-energy-weighted sum rule for the IVSGMR.
