Joyce’s orbifold construction and the twisted connected sums by Kovalev and Corti-Haskins-Nordström-Pacini provide many examples of compact Riemannian 7-manifolds with holonomy G₂ (i.e., G₂-manifolds). We would like to use this wealth of examples to guess further properties of G₂-manifolds and to find obstructions against holonomy G₂, taking into account the underlying topological G₂-structures. The Crowley-Nordström ν-invariant distinguishes topological G₂-structures. It vanishes for all twisted connected sums. By adding an extra twist to this construction, we show that the ν-invariant can assume all of its 48 possible values. This shows that G₂-bordism presents no obstruction against holonomy G₂. We also exhibit examples of 7-manifolds with disconnected G₂-moduli space. Our computation of the ν-invariants involves integration of the Bismut-Cheeger η-forms for families of tori, which can be done either by elementary hyperbolic geometry, or using modular properties of the Dedekind η-function.