Total intrinsic spin for plane gravity waves

A quantity which measures total intrinsic spin along the $z$ axis is constructed for planar gravity (fields dependent on $z$ and $t$ only), in both the Ashtekar complex connection formalism and in geometrodynamics. The total spin is conserved but (surprisingly) is not a surface term. This constant of the motion coincides with one of four observables previously discovered by Husain and Smolin. Two more of those observables can be interpreted physically as raising and lowering operators for total spin.