My Sister's Keeper starts with a prologue a few hundred words long. A nameless girl writes how, when she was three years old, tried to smother her sister with a pillow. However, her dad stopped her from going through with it. She writes how later in life she kept fantasizing about killing her sister, but the prologue ends with this statement:
In the end, though, I did not kill my sister. She did it all on her own. Or at least this is what I tell myself.
As a reader we don't even know any of the characters' names yet, but the author already spoils the ending. Somebody will die by the end of the book, either a murder or a suicide.
In chapter one we are introduced to Anna, who seeks the help of a lawyer to sue her parents. Anna's sister Kate has a rare form of cancer and her kidneys are failing and their parents want Anna to give up one of hers.
Throughout the story we learn Anna was born with a purpose. A genetically-screened test tube baby, the blood in her umbillical cord was used in an experimental therapy to extend Kate's life. Extend, not cure. And what started as a one-time sacrifice quickly becomes a regular occurrence. She's forced to give up blood and bone marrow, and has to take growth hormones even though she's not sick. Meanwhile she can't live a life of her own, and we, the reader, realize this circle of pain will keep going on until Kate dies.
The lawsuit unfolds, and we learn something new. Anna sued her parents on Kate's behalf. Instinctively we understand the events of the prologue are approaching rapidly. We will catch up with the here and now soon enough, the consequence chiseled in stone. Anna will win, gaining a life of her own at a terrible cost.
On the way to the hospital with her lawyer, they are rammed by a car and Anna dies. Kate gets both of Anna's kidneys, and in the epilogue we learn it's Kate who wrote the prologue.
The twist ending is maybe ten pages long, and although the story would suffer thematically, you could cut off the end and still have a functioning book. In fact, the author could have signed the prologue with Kate's name and it wouldn't have mattered. Why? Because we don't read to get to the conclusion. Rather than the suspense of 'how will this end,' the book's true strength is the palpable tension which exists between all of the characters.
Anna struggles against her parents for bodily independence, for the right to have her concerns be heard. She struggles against herself, torn between not wanting Kate to die and respecting her wishes. Between doing what's right and keeping the family from burning down. Her parents struggle with feelings of powerlessness, of clinging to false hope, the realization they've neglected their other children. The book's conclusion has no influence on any of these sources of tension.
Yes. You can start a story by giving away the ending, given you substitute mystery with tension. I think My Sister's Keeper is still a compelling read, if you can't tell by my screenname.