Following the approach of Haiden–Katzarkov–Kontsevich (Publ Math Inst Hautes Études Sci 126:247–318, 2017), to any homologically smooth mathbb {Z}-graded gentle algebra A we associate a triple (Sigma _A, Lambda _A; eta _A), where Sigma _A is an oriented smooth surface with non-empty boundary, Lambda _A is a set of stops on partial Sigma _A and eta _A is a line field on Sigma _A, such that the derived category of perfect dg-modules of A is equivalent to the partially wrapped Fukaya category of (Sigma _A, Lambda _A ;eta _A). Modifying arguments of Johnson and Kawazumi, we classify the orbit decomposition of the action of the (symplectic) mapping class group of Sigma _A on the homotopy classes of line fields. As a result we obtain a sufficient criterion for homologically smooth graded gentle algebras to be derived equivalent. Our criterion uses numerical invariants generalizing those given by Avella–Alaminos–Geiss in Avella et al. (J Pure Appl Algebra 212(1):228–243, 2008), as well as some other numerical invariants. As an application, we find many new cases when the AAG-invariants determine the derived Morita class. As another application, we establish some derived equivalences between the stacky nodal curves considered in Lekili and Polishchuk (J Topology 11:615–444, 2018)