The laws of physics are symmetrical, and if the future can be known perfectly given the past, so too, the past can be perfectly known given the future.

IF.

Which is the key to the matter. Whether the future is defined by the past is irrelevant - what matters is whether the nature of that determination can be known. And it can't.

So your 'if' clause fails, because it posits an impossible event. The future cannot be known.

And that's the key to understanding why we speak of choice. We can easily comprehend how one of our machines works, and we easily see that given knowledge of its states we can be highly accurate in predicting how it will change with time. But human minds are too complex for us to do this - so we attribute its operations, which we cannot comprehend, to chance. Ignorance is the key.