For a finite set D of nodes let E2(D)={(x, y)[mid ]x, y∈D, x≠y}. We define an inversive Δ2-structure g as a function g[ratio ]E2(D)→Δ into a given group Δ satisfying the property g(x, y)=g(y, x)−1 for all (x, y)∈E2(D). For each function (selector) σ[ratio ]D→Δ there is a corresponding inversive Δ2-structure gσ defined by gσ(x, y)=σ(x)·g(x, y)·σ(y)−1. A function η mapping each g into the group Δ is called an invariant if η(gσ)=η(g) for all g and σ. We study the group of free invariants η of inversive Δ2-structures, where η is defined by a word from the free monoid with involution generated by the set E2(D). In particular, if Δ is abelian, the group of free invariants is generated by triangle words of the form (x0, x1)(x1, x2)(x2, x0).