  On a smooth manifold with distributions ${\cal D}_1$ and ${\cal D}_2$ having
trivial intersection, we consider the integral of their mutual curvature, as a
functional of Riemannian metrics that make the distributions orthogonal. The
mutual curvature is defined as the sum of sectional curvatures of planes
spanned by all pairs of vectors from an orthonormal basis, such that one vector
of the pair belongs to ${\cal D}_1$ and the second vector belongs to ${\cal
D}_2$. As such, it interpolates between the sectional curvature of a plane
field (if both distributions are one-dimensional), and the mixed scalar
curvature of a Riemannian almost product structure (if both distributions
together span the tangent bundle). We derive Euler-Lagrange equations for the
functional, formulated in terms of extrinsic geometry of distributions, i.e.,
their second fundamental forms and integrability tensors. We give examples of
critical metrics for distributions defined on domains of Riemannian
submersions, twisted products and $f$-$K$-contact manifolds.
