We investigate dualities between Z₂ quotients of recently proposed compactifications of M-theory on `quantum geometries' of the form S¹∨S¹ and 10d orientifolds of type 0A and 0B string theories. In particular, we relate the Hořava-Witten theory on S¹∨S¹ to a 0B orientifold with gauge group SO(16)⁴. The resulting dictionary provides a geometric explanation for characteristic features of the 0B orientifold, such as the doubling of the gauge group, while the perturbative spectrum of the 0B orientifold indicates the emergence of novel M-theoretic degrees of freedom associated with the junction point. The 0B orientifold further reveals the existence of two variants of the theory on S¹∨S¹, corresponding to equal vs opposite (i.e., standard vs Fabinger-Hořava) orientations of the E₈ walls. We also analyze additional 0A and 0B orientifolds whose open string sectors do not arise from higher-dimensional gauge fields in M-theory and whose microscopic interpretation remains an open problem.