For many purposes in continuum mechanics it has been found useful to decompose the deformation gradient into two factors, which result from some elastic process through which destressing is achieved at a material point. There is an essential rotational non-uniqueness in these factors; however, some subfactors are uniquely defined. In particular, a certain unique right stretch tensor is identified, which serves as a convenient independent variable for describing the anisotropic elastic response of a solid from its evolving stress-free intermediate configurations.