It is known, by Gelfand theory, that every commutative JB * -triple admits a representation as a space of continuous functions of the formwhere L is a principal T-bundle and T denotes the unit circle in C. We provide a description of all orthogonality preserving (non-necessarily continuous) linear maps between commutative JB * -triples. We show that each linear orthogonality preserver T : C T 0 (L 1 ) → C T 0 (L 2 ) decomposes in three main parts on its image, on the first part as a positive-weighted composition operator, on the second part the points in L 2 where the image of T vanishes, and a third part formed by those points s in L 2 such that the evaluation mapping δs • T is non-continuous. Among the consequences of this representation, we obtain that every linear bijection preserving orthogonality between commutative JB * -triples is automatically continuous and biorthogonality preserving.