The map $f$ is a Teichmuller map with respect to a pair of quadratic differentials $q_1,q_1$ on $X$ --- initial differential and terminal differential. More precisely, $f$ maps the horisontal/vertical foliation of $q_1$ to the horisontal/vertical foliation of $q_2$.
To verify that $f$ is pseudo-anosov we need to constuct one pair of transversal measured foliations which is mapped by $f$ to itself. But $q_1$ and $q_2$ may not coincide up to isotopy, so their pairs of horisontal and vertical foliations will be different.