I don't know if that identity has a name. I brought it up to illustrate that
$$( {\vec a \times \nabla } ) \times \vec b \text{ and } ( {\nabla \times \vec a} ) \times \vec b$$
have different vectors being operated on while
$$( {\vec a \times \nabla } ) \times \vec b \text{ and } ( {\nabla \times \vec b} ) \times \vec a$$
have the same vector being operated on
These two forms arise in the gradient of dot product identity
$$\nabla_\vec{b} (\vec{a} \cdot \vec{b})=(\vec{a}\cdot \nabla)\vec{b}+\vec{a} \times (\nabla \times \vec{b})=\vec{a}(\nabla \cdot \vec{b})+(\vec{a} \times \nabla) \times \vec{b}$$
The nabla subscript b means that a is not operated on.
In your original question the key point was that one expression had a being operated on while the other had b being operated on. This is similar to in single variable calculus if we have
(D indicates the derivative)
D(uv)=uDv+vDu
uDv and vDu are not equal because
in uDv
v is being operated on while u is not
and
in vDu
u is being operated on while v is not
http://en.wikipedia.org/wiki/Vector_calculus_identities
Does not include the identity I mention, but it includes others that may be of interest.