  We show that while individual Riesz transforms are two weight norm stable
under biLipschitz change of variables on $A_{\infty}$ weights, they are two
weight norm unstable under even rotational change of variables on doubling
weights. More precisely, we show that individual Riesz transforms are unstable
under a set of rotations having full measure, which includes rotations
arbitrarily close to the identity. This provides an operator theoretic
distinction between $A_{\infty}$ weights and doubling weights.
  More generally, all iterated Riesz transforms of odd order are rotationally
unstable on pairs of doubling weights, thus demonstrating the need for
characterizations of iterated Riesz transform inequalities using testing
conditions for doubling measures, as opposed to the typically stable 'bump'
conditions.
