The adjoint method is widely used to compute parameter sensitivities in optimization and inverse problems involving time-dependent systems. However, solution of the adjoint problem typically necessitates the rematerialization of forward problem solution trajectories via checkpointing schemes, incurring additional storage, I/O, or recomputation costs that scale with problem size and duration. One proposed means of overcoming the performance limitations of conventional checkpointing schemes entails the use of reversible time integrators, enabling direct rematerialization of prior solution states via time-reversal of the evolution equations of the forward problem. Prior work has demonstrated the feasibility of implementing "bit-reversible" time integrators for dissipative dynamic systems which are capable of exactly reconstructing forward solution trajectories. However, the theoretical justification of such methods remains undeveloped. The work presented herein establishes a theoretical framework for developing and analyzing exactly bit-reversible time integrators for dissipative dynamic systems. Within this framework, irreversibility is assessed in terms of the rate of contraction of the solution state space, which is shown to be equal to the rate of information entropy production. To compensate for the apparent loss of information in the representation of distinguishable solution states, an "ancillary" solution state is introduced whose complementary evolution results in a measure-preserving transformation on the extended state space. Exactly bit-reversible implementations of such bijective mappings are achieved through the use of fixed-point arithmetic in combination with an entropy encoding procedure to ensure zero information loss in the numerical representation of discrete solution states. These concepts are extended to the development of exactly reversible time integrators for coupled systems of ODEs, with particular application to time-reversible visco-plastic constitutive models. Within this context the properties of bit-reversible integrators are further explored, including examinations of computational performance and relevant considerations for the efficient computation of adjoint sensitivities.