Calculating the variance of a family of tensors, each represented by a symmetric positive semi-definite second order tensor/matrix, involves the formation of a fourth order tensorRabcd. To form this tensor, the tensor product of each second order tensor with itself is formed, and these products are then summed, giving the tensorRabcdthe same symmetry properties as the elasticity tensor in continuum mechanics. This tensor has been studied with respect to many properties: representations, invariants, decomposition, the equivalence problem et cetera. In this paper we focus on the two-dimensional case where we give a set of invariants which ensures equivalence of two such fourth order tensorsRabcdandR~abcd. In terms of components, such an equivalence means that componentsRijklof the first tensor will transform into the componentsR~ijklof the second tensor for some change of the coordinate system.