Shear viscosities eta are reported for pure liquid methylcyclohexane (MCH) from 298.610 K to 333.694 K, for perfluoromethylcyclohexane (PFMCH) from 319.196 K to 333.114 K and for a MCH + PFMCH mixture of overall PFMCH near-critical mole fraction, x(c) = 0.3640, from (T-UCS/K-7) in the region of biphase liquid coexistence to (T-UCS/K + 20) in the uniphase region, where T-UCS = 320.13 K is the air-saturated upper liquid-liquid critical solution temperature. The measurements were made using a capillary rheometer that permits the measurement of the viscosity of thermally equilibrated coexisting-liquid phases. The results confirm that the near-critical viscosity exhibits a weak enhancement that strictly speaking becomes a divergence when account is taken of the finite shear gradients in the capillary during measurements. The viscosity of the uniphase mixture of critical composition is well-described after shear gradient correction by a multiplicative combination of an Arrhenius background and a critical power expression with an index close to the now-accepted universal value y = 0.0435. The chief objective of the work, in addition to contributing to knowledge of this aspect of near-critical rheology, is the development for the biphase of a simple expression for the temperature dependence of the viscosities of the coexisting phases, eta+ and eta-, that combines (a) an expression for the viscosity diameter 1/2(eta(+) + eta(-)) similar to that for the viscosity of the critical mixture in the uniphase region, with a similar best critical index y' between 0.041 and 0.0435, and (b) an expression for Delta eta (eta(+)-eta(-)) that behaves like an order parameter, with an index, very close to the normal value beta = 0.325 and as many Wegner correction terms as the data require. The best two-phase fit emerges from a freely fitted exponent y' approximate to 0.037 with one Wegner-extended scaling term, but we believe that were shear gradient correction to be applied, the best y' would be the consensus value y' = 0.0435. The magnitude of y' notwithstanding, we believe that our primary objective has been satisfied, namely, the formulation of an expression that affords a good description of the shear and background viscosities of near-critical mixtures in the one- and two-liquid phases in relation to our estimates of the nano- or molecular-viscosity derived from measurements of fluorescence polarization decay rates.