I consider the kinematical implications embedded in the decomposition of the deformation gradient F into two constituents, F* (say, elastic) and Fp (say, plastic), in the form F=F* Fp. Whatever the origin, nature, and physical meaning of these three deformation gradients, the decomposition can always be recast in the form F==V* Q Up, where V* and Up are stretch tensors, respectively associated with F* and Fp, and Q is a unique rotation tensor. It then follows that the (elastic) spin associated with V* and the (plastic) spin associated with Up are fully defined by their corresponding deformation rates. I give explicit expressions defining elastic and plastic spins in terms of the elastic and plastic deformation rates, respectively. I show that, if the total velocity gradient and either the elastic or the plastic (but not both) deformation rate tensor, are given, then all the remaining kinematical rate quantities can be calculated explicitly, with no constitutive or other assumptions. I give explicit expressions for the rate of change of V*, Q, and Up, from which these quantities can be incrementally but directly obtained. In particular, I give explicit expressions for the elastic spin, plastic spin, and the rigid spin. Constitutive implications of these purely kinematical results are briefly discussed in connection with rate-independent and rate-dependent phenomenological elastoplasticity, and, particularly, kinematical hardening.