Cyclic and symmetric polymorphisms and CSPs

The existence of symmetric polymorphisms of all arities has been shown recently, by Kun et al., to be equivalent to the existence of a set operation in a structure $\Gamma$, which was known to be equivalent to CSP($\Gamma$) being solved by a width-1 algorithm. We give an example of a relational structure with cyclic operations of all arities but no symmetric operations of all arities, and discuss its implications on the complexity of CSPs and related problems.