Abstract: Let X be a compact Riemann surface of genus g ≥ 2 and let G be the group Spin (8, ℂ) or PSO (8, ℂ). Let τ be a choice of a non-trivial outer automorphism of G of order 3, called triality. Suppose that X is equipped with a holomorphic order 3 automorphism α. Then α acts on the moduli space M (G) of principal G-bundles over X by taking the pull-back, and τ also acts on M (G), by modifying the action of G on the principal bundles through any automorphism of G of order 3 representing τ. The combination of these two actions gives an automorphism of order 3 of the moduli space M (G). In this work, the notion of α-trialitarian G-bundle is introduced to describe the fixed points of the automorphism of M (G) mentioned above. In particular, an equivalent notion of stability and polystability for such bundles is proved, and an explicit construction of them is provided.