  It is known, by Gelfand theory, that every commutative JB$^*$-triple admits a
representation as a space of continuous functions of the form
$$C_0^{\mathbb{T}}(L) = \{ a\in C_0(L) : a(\lambda t ) = \lambda a(t), \
\forall \lambda\in \mathbb{T}, t\in L\},$$ where $L$ is a principal
$\mathbb{T}$-bundle and $\mathbb{T}$ denotes the unit circle in $\mathbb{C}.$
We provide a description of all orthogonality preserving (non-necessarily
continuous) linear maps between commutative JB$^*$-triples. We show that each
linear orthogonality preserver $T: C_{0}^{\mathbb{T}} (L_1)\to
C_{0}^{\mathbb{T}} (L_2)$ 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 $\delta_s\circ T$ 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.
