Assume $M$ is a topological space and $f\in \operatorname{Homeo}(M)$, then $f$ obviously plays a significant role in the torus bundle

$$M_f = M\times I/\{(x,0)\sim (f(x),1)\mid x\in M\}.$$

Hence there should be some general result of the following type: $M_f$ and $M_g$ are bundle isomorphic (resp. diffeomorphic) if and only if "W", where W is a relation between $f$ and $g$.

Can someone help give W and explain? Thank you!