A classical transcendence result of Schneider on the modular $j$-invariant states that, for $\tau \in \mathbb{H}$, both $\tau$ and $j(\tau)$ are in $\overline{\mathbb{Q}}$ if and only if $\tau$ is contained in an imaginary quadratic extension of $\mathbb{Q}$. The space $\mathbb{H}$ has a natural interpretation as a parameter space for $\mathbb{Q}$-Hodge structures (or, in this case, elliptic curves), and from this perspective the imaginary quadratic points are distinguished as corresponding to objects with maximal symmetry. This result has been generalized by Cohen and Shiga-Wolfart to more general moduli of Hodge structures (corresponding to abelian-type Shimura varieties), and by Ullmo-Yafaev to higher dimensional loci of extra symmetry (special subvarieties), where bialgebraicity is intimately connected with the Pila-Zannier approach to the Andre-Oort conjecture. \\ \\ In this talk, I will discuss work in progress with Christian Klevdal on an analogous bialgebraicity characterization of special subvarieties in Scholze's local Shimura varieties and more general diamond moduli of $p$-adic Hodge structures.