how to construct a formula with an opposite SAT value of another one
The existence of such an algorithm would imply co-NP is included in NP which is unlikely

Checking that a formula is satisfiable: NP, generate proof by providing a solution

Checking that a formula is unsafisfiable: co-NP, generating proof can be harder, in classical proof systems the shortest proof of unsatisfiability of an arbitrary formula can be exponentially long 

The algorithm you are looking for would mean all problems in co-NP have a short refutation, by providing a solution to the "opposite" formula. Therefore we are unaware of the existence of such an algorithm.