In this paper, the derivation of irreducible bases for a class of isotropic 2nth-order tensors having particular "minor symmetries" is presented. The methodology used for obtaining these bases consists in extending the concept of deviatoric and spherical parts, commonly used for 2nd-order tensors, to the case of a nth-order tensor. It is shown that those bases are useful for effecting the classical tensorial operations and specially the inversion of a 2nth-order tensor. Finally, the formalism introduced in this study is applied for obtaining the closed form expression of the strain field within a spherical inclusion embedded in an infinite elastic matrix and subjected to linear or quadratic polynomial remote strain fields.