  In this study, we further the thermodynamic bootstrap program which involves
a set of recently developed ideas used to determine thermodynamic form factors
of local operators in integrable quantum field theories. These form factors are
essential building blocks for dynamic correlation functions at finite
temperatures or non-equilibrium stationary states. In this work we extend this
program in three ways. Firstly, we demonstrate that the conjectured
annihilation pole axiom is valid in the low energy particle-hole excitations.
Secondly, we introduce a crossing relation, which establishes a connection
between form factors with different excitation content. Typically, the crossing
relation is a consequence of Lorentz invariance, but due to the finite energy
density of the considered states, Lorentz invariance is broken. Nonetheless a
crossing relation involving excitations with both particles and holes can
established using the finite volume representation of the thermodynamic form
factors. Finally, we demonstrate that the thermodynamic form factors satisfy a
reparameterization invariance, an invariance which encompasses crossing.
Reparameterization invariance exploits the fact that the details of the
representation of the thermodynamic state are unimportant. In the course of
developing these results, we demonstrate the internal consistency of the
thermodynamic form factor bootstrap program in a number of ways. Finally, we
provide explicit computations of form factors of conserved charges and
densities with crossed excitations and show our results can be used to infer
information about thermodynamic form factors in the Lieb-Liniger model.
