Is there a polynomial-time reduction from a NP-hard problem to the complement of tautology?
The statement is: If a problem $P$ is $\mathbf{NP}$-hard, then there is a reduction from $\text{FALSIFIABLE}$ to $P$. ($\text{FALSIFIABLE}$ being the set of formulas for which there is an assignment which makes the formula false; this is trivially equal to the complement of $\text{TAUT}$.)

This statement is correct.

Why? Simply because $\text{FALSIFIABLE}$ is $\mathbf{NP}$-complete. You can prove this exactly the same way $\text{SAT}$ is proven to be $\mathbf{NP}$-complete, only inverting the truth values "true" and "false" (i.e., instead of searching for an assignment which makes the formula true, you search for an assignment which makes the formula false). This is not at all surprising; it is basically the duality principle of Boolean algebra.