When should твоих be used and when should своих be used? What are the reasons for picking one over another?
I think that the difference is the following:
"мой, твой, его (её, их)" denote "belonging to the speaker, the addressee, a third person".
"свой" is used when you want to denote "belonging to the subject of the sentence".
If the subject of the sentence is 1st or 2nd person pronoun, "свой" is preferred. In other cases, "мой", "твой", "его (её, их)" is used. It is not a strict rule, though.
If the subject is a quantification pronoun, than "свой" is the only option.
Please note that the subject is not always a noun in nominative case, it is a more complex notion. It also includes:
If the sentence is imperative, "свой" is the only option.
I do believe that Olga gave a succinct and accurate explanation. Just want to add that actually there are cases when
Let's, once again, refer our Bible )
Д. Э. Розенталь, "Литературное редактирование текста", "Возвратные и притяжательные местоимения".
To translate, this pronouns can be synonymous, especially when we can suspect a logical opposition - Touch my head with your hand.