Time-reversal invariance and charge symmetry have been studied in polarization measurements of single-nucleon transfer reactions in the mass five system. First, we experimentally established the equality of the proton analyzing power, $A$, for $^{4}\mathrm{He}$($\stackrel{\ensuremath{\rightarrow}}{p}$,$d$)$^{3}\mathrm{He}$ at 32 MeV and the polarization, $P$, of the inverse reaction, thereby checking time-reversal invariance. Then in a double scattering experiment using the reactions $^{3}\mathrm{H}$($d$,$\stackrel{\ensuremath{\rightarrow}}{n}$)$^{4}\mathrm{He}$ and $^{4}\mathrm{He}$($\stackrel{\ensuremath{\rightarrow}}{n}$,$d$)$^{3}\mathrm{H}$ at ${\ensuremath{\theta}}_{\mathrm{c}.\mathrm{m}.}=38.6\ifmmode^\circ\else\textdegree\fi{}$ and assuming $P=A$ (from time-reversal invariance) we obtained a polarization of 0.480 \ifmmode\pm\else\textpm\fi{} 0.016 for the 50 MeV neutrons. When this value is used, good agreement is found between the analyzing power distributions of $^{4}\mathrm{He}$($\stackrel{\ensuremath{\rightarrow}}{n}$,$d$)$^{3}\mathrm{H}$ and $^{4}\mathrm{He}$($\stackrel{\ensuremath{\rightarrow}}{p}$,$d$)$^{3}\mathrm{He}$ at 50 MeV, which is consistent with charge symmetry.NUCLEAR REACTIONS $^{4}\mathrm{He}$($\stackrel{\ensuremath{\rightarrow}}{p}$,$d$)$^{3}\mathrm{He}$, ${E}_{p}=32 \mathrm{and} 50$ MeV; $^{4}\mathrm{He}$($\stackrel{\ensuremath{\rightarrow}}{n}$,$d$)$^{3}\mathrm{H}$, ${E}_{n}=50$ MeV. Measured ${A}_{\ensuremath{\nu}}(\ensuremath{\theta})$, $\ensuremath{\theta}=25\ifmmode^\circ\else\textdegree\fi{}\ensuremath{-}150\ifmmode^\circ\else\textdegree\fi{}$ c.m. Inferred neutron polarization for reaction $^{3}\mathrm{H}$($\mathrm{d}$,$\stackrel{\ensuremath{\rightarrow}}{n}$)$^{4}\mathrm{He}$ at ${E}_{d}=37.1$ MeV, $\ensuremath{\theta}=38.6\ifmmode^\circ\else\textdegree\fi{}$ c.m.