  Frustration in 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 generic 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 wavevectors, labeled as
'octagonal (prism)' state and '(distorted) cubic' states. When the 'octagonal
prism' state is stabilized, non-zero 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.
