I am currently in the process of putting together a short article titled Were There Mystery Rites for Demeter Chthonia in Hermione? (which does what it says on the tin), and I have been reading over some of the research I have done on other Demetrian cults with (potential) Underworld provenance. The vast majority of these come from - surprise surprise - our old friend Pausanias, and two of the more interesting ones related to Persephone's abduction.
Persephone's abduction is a pretty interesting story, and the traditions that run outside what's presented in the Homeric Hymn to Demeter are particularly so. Pausanias offers several different accounts that share common elements, and the main crossovers relate to the traditions associated with Phigaleia (Paus. 8.42) and Thelpousa (Paus. 8.25.)
In Phigaleia, the local version tells of Demeter being raped by Poseidon after Persephone's disappearance. She fell pregnant and had a second daughter - Despoina, 'the Mistress'. Mourning for Persephone and feeling rather wrathful, Demeter retreated into a cave, dressing herself in black. With Demeter's retreat came the inevitable death of vegetation, livestock, and eventually the population. Zeus sent the Moirai to convince Demeter to set aside her anger, and restore the fertility of the land. She did so, and the Phigalians declared her cave-hideaway a sacred site, erecting an unusual statue in her honour:
Pausanias mentions that this unusual statue was burned, around the start of the fifth century BCE. Some time after the loss there was another period of barreness, and the Phigalians went to Delphi to find out why. They were told to dedicate a second statue to Demeter:
Arkadian, Azanian acorn-eaters,
By addressing the Arkadians as 'acorn-eaters' the response resonates with their claim to be the most ancient Greek people - that is, they lived on acorns before Demeter gave them the gift of agriculture, which she had now rescinded because they failed to properly venerate her. There is a lot to say about this 'oracle' alone, but what I want to draw out here is how 'anyone of intelligence with a good memory' would know why the original wooden statue was made in such an unusual way - and the answer can be picked out of the oracle: 'stallion-mated Deo'.
Demeter was raped by Poseidon in another local version of Persephone's abduction as well. Here, in Thelpousa, Demeter tried to rebuff Poseidon by disguising herself as a mare. In response, he turned into a stallion and then raped her. She has two cult epithets here - 'Erinys' because of the fury and anger she felt following the violation, and 'Louisa' - because after she set aside her grief and anger she bathed in a local river.
The horse plays a significant role in the Thelpousan tradition, and its cryptic inclusion in the Phigalian tradition shows a connection (at least to Pausanias) between the two cities. In Thelpousa, Demeter and Poseidon's forced union produces two offspring - the horse Arion and a daughter, whose name cannot be known to the uninitiated. Following the Phigalian tradition, Pausanias calls her Depsoina, and says she was the most worshipped divinity in the region. A daughter of Demeter, with a descriptive pseudonym, whose true identity is hidden to the uninitiated is a familiar trope, but Pausanias quite clearly differentiates between Despoina and Kore (who is unequivocally equated with Persephone by way of Homeric revelation), but he does not give Despoina a(nother) name. Persephone is directly relevant to Despoina's birth: her disappearance facilitates Demeter's rape.
Assuming that a reader of Pausanias's work will start at the beginning and read though to the end, they will have read the Thelpousan tradition, involving the horse-disguises, first, so they would know about this before getting to the Phigalian version and the oracle. This presents an interesting flattery into the text: a reader, who quite recently read the Thelpousan account, will recognise the horsey trope in the Phigalian account and count themselves in the intelligent and good-memoried class of people who understand exactly why the horse is being employed at Phigalia and the link to Telpousa. The stories are told far enough apart that it won't be immediately apparent to the reader why they remember the horse-related story of Demeter and Poseidon.
Or, perhaps I am giving Pausanias a little too much credit...