Classical pseudo-gauge transformations are discussed in the context of hydrodynamic models of heavy-ion collisions. A decomposition of the pseudo-gauge transformation into Lorentz-invariant tensors is made, which allows for better interpretation of its physical consequences. For pseudo-gauge transformations connecting two symmetric energy-momentum tensors, we find that the super-potential must obey a conservation law of the form = 0. This equation, referred to below as the STS condition, represents a constraint that is hardly possible to be satisfied for tensors constructed out of the basic hydrodynamic variables such as temperature, baryon chemical potential, and the hydrodynamic flow. However, in a special case of the boost-invariant flow, the STS condition is automatically fulfilled and a non-trivial residual pseudo-gauge transformation defined by a single scalar field is allowed. In this case the bulk and shear viscosity coefficients become pseudo-gauge dependent; however, their specific linear combination appearing in the equations of motion remains pseudo-gauge invariant. This finding provides new insights into the role of pseudo-gauge transformations and pseudo-gauge invariance.