This work is intended to provide a convenient tool for the mathematical analysis of a particular kind of finite volume approximations which can be used, for instance, in the context of nonlinear and/or anisotropic diffusion operators. Following the approach developed by F.~Hermeline and by K.~Domelevo and P.~Omnès, we consider a ``double'' covering $\Tau$ of a three-dimensional domain by a rather general primal mesh and by a well-chosen ``dual'' mesh. The associated discrete divergence operator $\mbox{\rm div}^{\Tau}$ is obtained by the standard finite volume approach. Then a consistent discrete gradient operator $\nabla^\Tau$ is defined in such a way that $-\mbox{\rm div}^{\Tau}$, $\nabla^\Tau$ enjoy an analogue of the integration-by-parts formula known as the ``discrete duality property''. We discuss the implications of these properties and give a brief survey of other ``discrete calculus'' tools for ``double'' finite volume schemes.