A short survey on the experimental testing of solid propellants has highlighted finite strain responses that are temperature-dependent, viscoelastic with damage, and exhibit tension/compression asymmetry. Consequently, a finite strain viscoelastic model that satisfies the principles of thermodynamics has been developed. This model is based on the common multiplicative decomposition of the deformation gradient into elastic and viscous components, with considerations for damage and asymmetry. The model has been tested against three sets of data from the literature, carefully selected to represent the various characteristics of solid propellants. The model accurately reproduces uniaxial tension responses at different strain rates and temperatures, with the capability to account for superimposed hydrostatic pressure. Notably, these satisfactory representations require only five fitting parameters, in addition to the typical identification of polymer linear viscoelasticity and timetemperature superposition. Finally, an attempt to reproduce both tension and compression tests conducted independently on the same material underscores the need to account for tension-compression asymmetry, as defined in the proposed constitutive equations. This finding advocates for new tests, such as compression following tension and vice versa.