Suppose we have theories $T$ and $H$ that are bi-interpretable, now suppose that the relevant interpretations achieving that bi-interpretability are: $\tau: T \to H$, and $\pi: H \to T$. Now suppose that the domain of $\tau$ is a proper subdomain of theory $H$, and suppose the domain of $\pi$ is the whole domain of theory $T$. Suppose that $T$ and $H$ are not synonymous.
Is it always the case that there cannot be bi-interpretability achieving interpretations that has the reverse sub-domain, whole domain relationship, i.e. there cannot be bi-interpretability achieving interpretations $\tau_1: T \to H$, and $\pi_1: H \to T$ such that the domain of $\tau_1$ is the whole domain of theory $H$, and the domain of $\pi_1$ is a proper sub-domain of theory $T$?
Is synonymy relevant to this matter? Like saying if $T$ and $H$ were synonymous, then we can have the reverse state mentioned above?
