Introduction to Foliations and Lie Groupoids, Book by Ieke Moerdijk and Janez Mrčun, Chapter 1, Page 16.
The smooth action of $\mathbb{Z}$, defined on $\mathbb{R} \times F$ by $$(k,(t,x)) \mapsto (t+k, f^k(x))$$ or $$(k,(t,x)) \mapsto (t-k, f^k(x))$$
The first choice in the one that is mentioned in the book but I think the second choice is the right one.
As I alluded to in the comments, these two group actions are equivalent. That is, if $\Psi_1$ and $\Psi_2$ denote the two group actions, there is a diffeomorphism $\varphi : \mathbb{R}\times F \to \mathbb{R}\times F$, $(t, x) \mapsto (-t, x)$ such that the following diagram commutes
$$\require{AMScd} \begin{CD} \mathbb{R}\times F @>{\varphi}>> \mathbb{R}\times F\\ @V{\Psi_1}VV @VV{\Psi_2}V \\ \mathbb{R}\times F @>{\varphi}>> \mathbb{R}\times F. \end{CD}$$
Said another way, the two group actions are conjugate via $\varphi$; i.e. $\Psi_2 = \varphi\circ\Psi_1\circ\varphi^{-1}$. It follows that $\varphi$ descends to a diffeomorphism of the quotients. In the first case, the quotient is the mapping torus of $f$, and in the second case, the quotient is the mapping torus of $f^{-1}$.