Frustration in a spin system can give rise to unique ordered states and as a consequence several physical phenomena are expected, such as multiferroics, high temperature superconductors, and anomalous Hall effect. Here we report the "new magnetic orders" induced by anisotropic spin exchanges on pyrochlore spinels as the interplay of spin-orbit coupling and geometrical frustration. Due to complicated superexchange paths of B-site spinels, we claim that anisotropic interaction between next-nearest neighbors play an important role. Based on the systematic studies of the spin model, we argue that several classical spin states can be explored in spinel systems; local XY state, all-in all-out state, Palmer-Chalker state, and coplanar spiral state. In addition, we reveal new types of magnetic phases with finite ordering wave vectors, labeled as octagonal (prism) state and (distorted) cubic states. When the octagonal prism state is stabilized, nonzero scalar spin chirality induces alternating net current in addition to finite orbital current and orbital magnetization even in Mott insulators. Finally, we also discuss the relevance of distorted cubic state to the magnetic order of spinel compound GeCo2O4.