  We present a new mimetic finite difference method for diffusion problems that
converges on grids with \textit{curved} (i.e., non-planar) faces. Crucially, it
gives a symmetric discrete problem that uses only one discrete unknown per
curved face. The principle at the core of our construction is to abandon the
standard definition of local consistency of mimetic finite difference methods.
Instead, we exploit the novel and global concept of $P_{0}$-consistency.
Numerical examples confirm the consistency and the optimal convergence rate of
the proposed mimetic method for cubic grids with randomly perturbed nodes as
well as grids with curved boundaries.
