Unknown title for old science fiction story
I'm having trouble finding an old book, I remember the basic plot, but I can't remember the title. I'm hoping someone has a better clue than it.
In short the book is about a young girl/woman who can see and manipulate "rings" that can be used to pass through to alternate worlds. The two main worlds are a rocky hot world in which travelers become furry four legged beasts, the other is one is an ocean world where the travelers become small aquatic creatures. The plot revolves around her escape from her prison as well as the discovery of her past including "doubled rings" and some alternate time selves as well as the discovery that her captor, her local bully, and an elderly gentleman are all the same person duplicated with the double rings. I can't remember the names of the characters, the author or title, so I'd really appreciate it if anyone knew the title of the book.