This work is dedicated to the study of the Möbius invariant class of constrained Willmore surfaces and its symmetries. We define a spectral deformation by the action of a loop of flat metric connections; Bäcklund transformations, by applying a dressing action; and, in 4-space, Darboux transformations, based on the solution of a Riccati equation. We establish a permutability between spectral deformation and Bäcklund transformation and prove that non-trivial Darboux transformation of constrained Willmore surfaces in 4-space can be obtained as a particular case of Bäcklund transformation. All these transformations corresponding to the zero multiplier preserve the class of Willmore surfaces. We verify that, for special choices of parameters, both spectral deformation and Bäcklund transformation preserve the class of constrained Willmore surfaces admitting a conserved quantity, and, in particular, the class of CMC surfaces in 3-dimensional space-form.