Gradings of matrix algebras by the Klein group

We classify the C-2 x C-2-gradings on M-2(k), k an arbitrary field. If char(k) not equal 2 our approach relies on the duality between gradings and actions for finite abelian groups, and if k is algebraically closed we find precisely one isornorphism type of grading which is not isomorphic to a grading with all the matrix units being homogeneous elements. If char(k) = 2 we use a computational approach and we find that any C-2 x C-2-grading is induced by a C-2-grading.