International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2015 
International Tables for Crystallography (2015). Vol. A, ch. 1.5, p. 83

If the direct or crystal basis is transformed by the transformation matrix P: , the corresponding basis vectors of the reciprocal (or dual) basis transform as (cf. Section 1.3.2.5 )where the notation is applied (cf. Section 1.5.1.2).
The quantities that transform in the same way as the basis vectors are called covariant with respect to the basis and contravariant with respect to the reciprocal basis . Such quantities are the Miller indices (hkl) of a plane (or a set of planes) in direct space and the vector coefficients (h, k, l) of the vector perpendicular to those planes, referred to the reciprocal basis :Quantities like the vector coefficients of any vector in direct space (or the indices of a direction in direct space) are covariant with respect to the basis vectors and contravariant with respect to :