The local-global property for G-invariant terms

For some Maltsev conditions Σ it is enough to check if a finite algebra A satisfies Σ locally on subsets of bounded size, in order to decide, whether A satisfies Σ (globally). This local-global property is the main known source of tractability results for deciding Maltsev conditions. In this paper we investigate the local-global property for the existence of a G-term, i.e. an n-ary term that is invariant under permuting its variables according to a permutation group G ≤ Sym(n). Our results imply in particular that all cyclic loop conditions (in the sense of Bodirsky, Starke, and Vucaj) have the local-global property (and thus can be decided in polynomial time), while symmetric terms of arity n > 2 fail to have it.