We discuss the perturbative expansion of SU(N) Yang-Mills theories defined on a d-dimensional torus of linear size l with twisted boundary conditions, generalizing previous results in the literature. For a specific class of twist tensors depending on a single integer flux value k, we show that perturbative results to all orders depend on the combination lN 2/d and a flux-dependent angleθ . This implies a new kind of volume independence that holds at finite N and for fixed values ofθ . Our

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... s also provide interesting information about the possible occurrence of tachyonic instabilities at one-loop order. We support the prescription that instabilities are avoided, if the large N limit is taken keepingθ >θ c , and appropriately scaling the magnetic flux k with N. Numerical results in 2+1 dimensions provide a test of how these ideas extend into the non-perturbative regime.