  We construct a family of hyperbolic link complements by gluing tangles along
totally geodesic four-punctured spheres, then investigate the commensurability
relation among its members. Those with different volume are incommensurable,
distinguished by their scissors congruence classes. Mutation produces
arbitrarily large finite subfamilies of nonisometric manifolds with the same
volume and scissors congruence class. Depending on the choice of mutation,
these manifolds may be commensurable or incommensurable, distinguished in the
latter case by cusp parameters. All have trace field Q(i,\sqrt{2}), but some
have integral traces while others do not.
