  The present knowledge of Lamb shift, fine-, and hyperfine structure of the 2S
and 2P states in muonic helium-3 ions is reviewed in anticipation of the
results of a first measurement of several $\mathrm{2S\rightarrow2P}$ transition
frequencies in the muonic helium-3 ion, $\mathrm{\mu^3He^+}$. This ion is the
bound state of a single negative muon $\mu^-$ and a bare helium-3 nucleus
(helion), $\mathrm{^3He^{++}}$.
  A term-by-term comparison of all available sources, including new, updated,
and so far unpublished calculations, reveals reliable values and uncertainties
of the QED and nuclear structure-dependent contributions to the Lamb shift and
the hyperfine splitting. These values are essential for the determination of
the helion rms charge radius and the nuclear structure effects to the hyperfine
splitting in $\mathrm{\mu^3He^+}$. With this review we continue our series of
theory summaries in light muonic atoms; see Antognini et al., Ann. Phys. 331,
127 (2013), Krauth et al., Ann.Phys. 366, 168 (2016), and Diepold et al., ArXiv
1606.05231 (2016).
