I feel like I kind of know this, but sometimes I feel unsure. It's pretty similar to the Swedish sin, sitt. Is there a good rule or a nice, short explanation for when and where to use si, sia instead of li, sxi?
Use si or sia when referring to the already-introduced subject of the sentence:
- La virino rigardis la bildon antaŭ si -> "The woman looked at the picture in front of her", the picture is in front of the subject of the sentence ("the woman")
- Li eniris sian domon -> "He entered his (own) house", the house belongs to the subject of the sentence ("he")
- La infano amas sian patrinon -> "The child loves its (own) mother", the mother belongs to the subject of the sentence ("the child")
You asked for short:
Use si/sia when both are true
- the pronoun is not part of the subject.
- the pronoun refers to the subject of the verb.
Regarding point 1: you'll never see la viro kaj sia edzino estas... because lia edzino is part of the subject.
Regarding point 2: it's important to notice that it's the subject of the verb and not the subject of the sentence, because while they're usually the same, they are not always the same.
- Mi vidis lin bati sin (the guy hit himself.)
The grammar at Lernu regarding pronouns has detailed information about the use of si and sia, available in many languages (the link is to the English version), that is too long to copy here with all the examples. It is worth checking it out.