Can valid operations applied on an incorrect premises ever lead to true conclusions? When we do a proof by contradiction, we assume a premise, derive a result from it, and if the result is incorrect then we conclude that our initial premise was wrong. However, if we do get a correct result, can we conclude that the premise is true, or was out test just inconclusive?
My guess is that we can go ahead and assume that the promise was correct. If it was possible to derive a true result from a false premise using valid operations, then it would also be possible to work backwards from a correct result to a false premise by performing valid operations. However, something just tells me that this is "too good to be true".   
 A: You can certainly derive true statements from false premises. By the principle of explosion, a false premise implies every statement.
To be concrete, assume $-1 = 1$ (obviously false). Squaring both sides, we get $1 = 1$ (obviously true). Here the problem is that squaring isn't an "operation" that you can undo.
A: It is possible to begin from false assumptions, perform valid operations, and arrive at a true statement. For instance, assume $0=1$. Multiplying both sides by $0$ yields the true statement $0=0$.
For this reason, you cannot prove that a statement is true by assuming that it is true and then deriving something else that is true. This is a common mistake I've seen among calculus students. Often they just need to write their equations in a different order, and everything works out because the operations they're doing are reversible, but the logical order of statements is important.
You say:

If it was possible to derive a true result from a false premise using valid operations, then it would also be possible to work backwards from a correct result to a false premise by performing valid operations.

This statement is untrue exactly for the reason that not every valid operation is reversible. For instance, in my example we cannot reverse multiplication by $0$ by dividing by $0$. It's not a valid operation. If you're familiar with dot products, then you know that dotting by a vector is not reversible - i.e., if we know $u \cdot w = v\cdot w$, we cannot undo dotting with $w$ to obtain $u=v$. There are tons more examples of operations we do that are not reversible, but this should give you the idea. 
