  We study horizontal subvarieties $Z$ of a Griffiths period domain $\mathbb
D$. If $Z$ is defined by algebraic equations, and if $Z$ is also invariant
under a large discrete subgroup in an appropriate sense, we prove that $Z$ is a
Hermitian symmetric domain $\mathcal D$, embedded via a totally geodesic
embedding in $\mathbb D$. Next we discuss the case when $Z$ is in addition of
Calabi-Yau type. We classify the possible VHS of Calabi-Yau type parametrized
by Hermitian symmetric domains $\mathcal D$ and show that they are essentially
those found by Gross and Sheng-Zuo, up to taking factors of symmetric powers
and certain shift operations. In the weight three case, we explicitly describe
the embedding $Z\hookrightarrow \mathbb D$ from the perspective of Griffiths
transversality and relate this description to the Harish-Chandra realization of
$\mathcal D$ and to the Kor\'anyi-Wolf tube domain description. There are
further connections to homogeneous Legendrian varieties and the four Severi
varieties of Zak.
