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