The final state of turbulent magnetic relaxation in a reversed field pinch is well explained by Taylor's hypothesis. However, recent resistive-magnetohydrodynamic simulations of the relaxation of braided solar coronal loops have led to relaxed fields far from the Taylor state, despite the conservation of helicity. We point out the existence of an additional topological invariant in any flux tube with a nonzero field: the topological degree of the field line mapping. We conjecture that this constrains the relaxation, explaining why only one of three example simulations reaches the Taylor state.