Local-global property for G-invariant terms

For some Maltsev conditions Sigma it is enough to check if a finite algebra A satisfies Sigma locally on subsets of bounded size in order to decide whether A satisfies Sigma (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.