Owing to their chiral cubic structure, exotic multifold topological excitations have been predicted and recently observed in transition metal silicides like β-RhSi. Herein, we report that the topological character of RhSi is also observed in its orthorhombic α-phase which displays multiple types of Dirac nodes very close to the Fermi level (εF ) 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 εF . 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 εF around and near the S-point on the ky − kz 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.