May siblings of the opposite sex that share only one parent be secluded (yichud) with one another? Does it matter which parent they share?
Nitei Gavriel (Hilchot Yichud, chapter 3, halacha 11, footnote 14) brings several opinions (including R' Moshe) that all half-siblings have the same halachot as full siblings. This is what the Nitei Gavriel says in the halacha as well.
He then brings that R' Eliyashev says only half-siblings on the mother's side are considered like full siblings. The reason given is that they have a natural feeling of love to one another (that I'm assuming is more than the natural feeling of love half-siblings on the father's side share).
The Nitei Gavriel (I think) then concludes that it really only applies if the half-siblings grew up together (I think he is saying all the opinions hold this). My understanding is that if they have not grown up together than there is no familial bond felt between them, and therefore they are treated as strangers with regards to Yichud.
See Beis Shmuel, Even HoEzer 22:1. The permissibility of Yichud of a brother and sister includes half-brothers and half-sisters — Dvar Halachah Hosafos Chadashos 2; Divrei Sofrim Emek Davar 56. However, there is a prohibition of Yichud with a step-brother or step-sister.