Summary: The resultant two-dimensional (2D) balance laws of mass, linear and angular momentum, and energy as well as the entropy inequality for shells are derived by direct through-the-thickness integration of corresponding 3D laws of continuum thermomechanics. It is indicated that the resultant shell stress power cannot be expressed exactly through the 2D shell stress and strain measures alone. Hence, an additional stress power called an interstitial working is added to the resultant 2D balance of energy. The new, refined, resultant balance of energy and entropy inequality derived here are regarded to be exact implications of corresponding global 3D laws of rational thermodynamics. The kinematic structure of our shell theory is that of the Cosserat surface, while our refined resultant laws of thermomechanics contain three additional surface fields somewhat similar to those present in 3D extended thermodynamics. We briefly analyse the restrictions imposed by our refined resultant entropy inequality on the forms of 2D constitutive equations of viscous shells with heat conduction and of thermoelastic shells. It is shown, in particular, that in such shells the refined resultant entropy inequality allows one to account for some longer-range spatial interactions. We also present several novel forms of 2D kinetic constitutive equations compatible with the resultant shell equations.