The Torelli locus - the image of the moduli space of curves (Mg) in the moduli space of abelian varieties (Ag) - is much-studied but still mysterious. In characteristic p, Ag has a beautiful stratification by the isomorphism type of A[p], and examples show that Mg is far from transverse to this stratification. In an ongoing project, we develop tools to understand (and perhaps make principled conjectures about) which strata of Ag meet Mg. In this talk, we explain some of the structures involved and give new results about them. Parts of this are joint work with Bryden Cais and Rachel Pries.