Learn Russian: Анна умывалась и собирала свои вещи. - Anna was washing her face and packing her things.

The reflexive possessive pronoun свои refers back to the subject of the sentence, Anna. It shows she is packing her own things. If you used её, a Russian speaker would assume Anna was packing some other woman's things!

What case is the phrase свои вещи in this sentence?

It is in the Accusative case because it is the direct object of the action собирала (was packing). For plural inanimate objects like вещи (things), the Accusative case form looks exactly the same as the Nominative (dictionary) form.

Can свои be used with subjects like 'I' or 'they'?

Yes! Unlike English, which changes the word (my own, their own), the root свой works for any subject as long as the subject owns the object. If the sentence was 'I am packing my things', you would still use свои вещи.

Why does умывалась end in -сь instead of the usual reflexive ending -ся?

The reflexive suffix is -ся after consonants, but it shrinks to -сь after vowels. Since the past tense feminine verb ends in the vowel -а (умывала), we attach -сь to get умывалась (she was washing herself).