A paired cranked Nilsson-Strutinsky-Bogoliubov (CNSB) model is presented, which employs the same method to calculate the liquid-drop energy and moment of inertia as the unpaired cranked Nilsson-Strutinsky (CNS) model. In the CNSB model, the energy minimization is carried out in the mesh of pairing gaps and Fermi levels λ as well as deformation parameters. The high spin states in 161 Lu are then investigated with the CNSB and CNS models. The terminating structure shows a striking similarity with these two models. Combining the CNSB and CNS models, a complete understanding of high spin structures, including the normal deformed (ND) and triaxial strongly deformed (TSD) bands and observed side bands in 161 Lu, is achieved. It appears that the only important paired crossings are the first i 13/2 neutron crossing and the first h 11/2 proton crossing. For the description of the unpaired high spin crossings, it is important to be able to distinguish between the pseudospin partners in the proton N = 4 shell, (d 5/2 ,g 7/2 ) and (d 3/2 ,s 1/2 ). The yrast bands are predicted to terminate, which explains the structure of a TSD-like band X2. A band crossing at I ≈ 36.5 for the TSD band in 161 Lu, unique within the chain of even-N Lu isotopes, is well described by the CNSB model.