In this paper, the extreme problem for Orlicz and L q torsional rigidity is discussed. Moreover, we introduce Orlicz and L q geominimal torsional rigidity, which is defined as being motivated through Orlicz L ϕ mixed torsional rigidity. Also, the invariance of Orlicz and L q geominimal torsional rigidity under orthogonal matrices is proved, and isoperimetric type inequality and circular type inequality for the torsional rigidity are established as well