  In this paper we study equivariant constrained Willmore tori in the 3-sphere.
These tori admit a 1-parameter group of M\"obius symmetries and are critical
points of the Willmore energy under conformal variations. We show that the
associated spectral curve of an equivariant torus is given by a double covering
of $\mathbb C$ and classify equivariant constrained Willmore tori by the genus
g of their spectral curve. In this case the spectral genus satisfies $g \leq
3.$
