Hi Dror, sorry, I didn't check this for awhile. The curves $X_t$ and $X_{1-t}$ have a meaning: they are a Drinfeld modular curve and a Drinfeld-Shimura curve, respectively. I thought it was cute that they had such a simple formula. The Jacquet-Langlands correspondence combined with the Tate conjecture for divisors on abelian varieties over function fields produces the desired isogeny. I wanted to know if there was an explicit correspondence, but maybe this is too much to hope for.

Thank you Xarles! I am attempting to see whether your construction generalizes. The Jacobian of $X_t$ is isogenous to a product of Jacobians of hyperelliptic curves $C_{1,t}$ and $C_{2,t}$. Evidence suggests that your pattern persists: the Jacobians of $C_{1,t}$ and $C_{1,1-t}$ are isogenous over $\mathbf{F}_{q^2}(t)$, and the Jacobians of $C_{2,t}$ and $C_{2,1-t}$ are isogenous already over $\mathbf{F}_q(t)$.

Thanks so much for this answer, and for the clear explanation using vector bundles, especially the explanation of the constant term map. In the Deligne-Flicker paper you cite, Prop. 7.1 seems to be saying that for $\mathrm{deg} D = 4$, the number of cusp forms is $q$ on the nose, is that correct?