This paper discusses the different approaches that can be used to determine the strain energy density of a given rubber-like material based on tension–torsion experimental results. More precisely, the aim is to answer the question: how to handle the measured macroscopic quantities, i.e. load and torque, to determine the constitutive equation with the less possible assumptions? The method initially proposed by Penn and Kearsley [Trans. Soc. Rheol. 20 (1976) 227–238] is adopted: the strain energy derivatives with respect to kinematical quantities have to be calculated in terms of the measured load and torque. Here, we propose to consider different sets of kinematical quantities to overcome the incoherence encountered with the classical Cauchy–Green strain invariants I1 and I2. Two new sets are considered: the principal stretch ratios and two specific invariants of the logarithmic (true) Hencky strain tensor. The corresponding derivations coupled with new experimental results permit (i) to calculate the Cauchy stress tensor on the outer surface of the cylindrical samples, and (ii) to demonstrate that a well- conditioned set of kinematical quantities must be adopted to determine the strain energy density. It is proved here that the principal stretch ratios are good candidates to express and determine the strain energy density with tension–torsion experiments.