A sufficient condition for a quantum state of a system of spin-1/2 particles to admit a local hidden variable description, i.e., to be classical, is the separability of the density matrix characterizing its state, but not all classical states are separable. This leads one to infer that separability and classicality are two different concepts. These concepts are examined here in the framework of a criterion for identifying the classicality of a system of spin-1/2 particles based on the concept of joint quasiprobability (JQP) for the eigenvalues of spin components [Puri, J. Phys. A 29, 5719 (1996)]. The said criterion identifies a state as classical if a suitably defined JQP of the eigenvalues of spin components in three suitably chosen orthogonal directions is non-negative. In agreement with other approaches, the JQP-based criterion also leads to the result that all nonfactorizable pure states of two spin-1/2 particles are nonclassical. In this paper it is shown that the application of the said criterion to mixed states suggests that the states it identifies as classical are also separable and that there exist states which, identified as classical by other methods, may not be identified as classical by the criterion as it stands. However, results in agreement with the known ones are obtained if the criterion is modified to identify as classical also those states for which the JQP of the eigenvalues of the spin components in two of the three prescribed orthogonal directions is non-negative. The validity of the modified criterion is confirmed by comparing its predictions with those arrived at by other methods when applied to several mixed states of two spin-1/2 particles and the Werner-like state of three spin-1/2 particles [T\`oth and Ac\`{\i}n, Phys. Rev. A 74, 030306(R) (2006)]. The JQP-based approach, formulated as it is along the lines of the
P-function approach for identifying classical states of the electromagnetic field, offers a unified approach for systems of an arbitrary number of spin-1/2 particles and the possibility of linking classicality with the nature of the measurement process.