The symmetric stress tensor (SST) has been widely accepted in classical fluid mechanics for nearly two centuries, particularly following the work of Stokes in 1845. Its acceptance has been reinforced by the successful application of the Navier–Stokes equations to many practical flow problems. Nevertheless, the author has argued that certain interpretations of the SST lead to inconsistencies that warrant further examination. Establishing and communicating this argument has proved difficult, particularly because the proposed interpretation challenges a formulation routinely taught in fluid mechanics. Earlier attempts to publish these arguments encountered objections, including the absence of direct experimental measurements that distinguish the competing descriptions of the relevant stress components. For example, a tangential wall shear stress can be measured, whereas the corresponding transverse component invoked in the author’s interpretation is not measured in the same manner. The present investigation therefore develops the argument through detailed examinations of familiar flow configurations, supported by a reported comparison between rotating water and rotating solid discs. The analysis considers six principal questions. First, in simple Couette flow, the author questions the interpretation of a transverse shear stress when the velocity has no component in that direction, relating this concern to the definition of a fluid as a material that deforms continuously under an applied shear stress. Second, the finite-element stress argument presented here is interpreted as implying uniform stresses under an imposed symmetry requirement, which would prevent the viscous force required in the momentum equations from being obtained. Third, conventional definitions of wall shear stress are examined to determine how the full stress tensor is represented in surface-force calculations. Fourth, the author’s proposed inclusion of an additional transverse stress in the Stokes-flow calculation preserves the sphere’s drag but produces a nonzero lift, contrary to the symmetry of the flow. Fifth, planar Poiseuille flow is used to examine an apparent conflict, within the proposed interpretation, between stress symmetry and the spatial increments required by differential calculus. Sixth, rigid rotation is examined to distinguish zero viscous dissipation from the transmission of mechanical energy under different loading conditions. A further discussion concerns the dissipation function for compressible flow under Stokes’ hypothesis. These arguments are presented as the author’s interpretation of the specified examples, rather than as experimentally established replacements for classical continuum mechanics. The purpose of the investigation is to clarify the physical meaning assigned to stress components and to encourage critical examination of the assumptions used in their construction. Comments on the derivations, their interpretation, and the experimental comparisons are welcomed.