Her aunt having left the room, I decided to read the book.
Leaving the room, he tripped over the mat.
Both leaving phrases are called present participle clauses in their grammar.
Covered with confusion, I left the room.
We left the room and went home, the job finished.
Covered phrase and job finished phrase are past participle phrases.
You can read Quirk's grammar book to know more.
Hope this helps.