Extending work of Kapouleas and Yang, we construct sequences of closed minimal surfaces embedded in the round unit 3-sphere and converging to a Cliﬀord torus counted with multiplicity. Each surface resembles multiple parallel copies of the limit torus connected by many small catenoidal tunnels positioned at the sites of rectangular lattices on the copies. The symmetries of the surface are precisely the isometries of the 3-sphere preserving the union of these lattices. In general this symmetry group is generated by reﬂections through certain spheres having equator an axis circle of the limit torus, but when the lattices are square the group also admits reﬂections through certain great circles on the torus. The collection of sequences is indexed by the number—any integer greater than 1—of tori glued together and by the common ratio—any positive rational—of the lattice edges. Each sequence itself is indexed by a further positive integer which determines the number of catenoids (and so the number of lattice sites). This last number is not arbitrarily prescribed but must be chosen large in terms of a constant depending on the other indices. The proof relies on singular perturbation methods, in particular the gluing methodology developed by Kapouleas.