Low-rank tensor recovery has been widely applied to computer vision and machine learning. Recently, tubal nuclear norm (TNN) based optimization is proposed with superior performance as compared to other tensor nuclear norms. However, one major limitation is its orientation sensitivity due to low-rankness strictly defined along tubal orientation and it cannot simultaneously model spectral low-rankness in multiple orientations. To this end, we introduce two new tensor norms called OITNN-O and OITNN-L to exploit multi-orientational spectral low-rankness for an arbitrary K-way (K ≥ 3) tensors. We further formulate two robust tensor decomposition models via the proposed norms and develop two algorithms as the solutions. Theoretically, we establish non-asymptotic error bounds which can predict the scaling behavior of the estimation error. Experiments on real-world datasets demonstrate the superiority and effectiveness of the proposed norms.