  Owing to their chiral cubic structure, exotic multifold topological
excitations have been predicted and recently observed in transition metal
silicides like $\beta$-RhSi. Herein, we report that the topological character
of RhSi is also observed in its orthorhombic $\alpha$-phase which displays
multiple types of Dirac nodes very close to the Fermi level ($\varepsilon_F$)
with the near absence of topologically trivial carriers. We discuss the
symmetry analysis, band connectivity along high-symmetry lines using group
representations, the band structure, and the nature of the Dirac points and
nodal lines occurring near $\varepsilon_F$. The de Haas-van Alphen effect
(dHvA) indicates a Fermi surface in agreement with the calculations. We find an
elliptically-shaped nodal line very close to $\varepsilon_F$ around and near
the $S$-point on the $k_y-k_z$ plane that results from the intersection of two
upside-down Dirac cones. The two Dirac points of the participating Kramers
degenerate bands are only 5 meV apart, hence an accessible magnetic field might
induce a crossing between the spin-up partner of the upper-Dirac cone and the
spin-down partner of the lower Dirac cone, possibly explaining the anomalies
observed in the magnetic torque.
