Lagrange, Hermite, and Birkhoff interpolation methods are all central to approximation theory, numerical analysis and for instance the generation of quadrature rules. In case it is desired to use these univariate methods in multiple dimensions, a tensor product approach is for example suitable for this extension. In this article, tensor product Birkhoff interpolation schemes are considered in their full generality. The interpolation points are equally spaced and they are forming an \(n\)-dimensional lattice. Here, the Birkhoff interpolants are sought by using Lagrange bases of the multidimensional polynomial space in order to reduce their computational complexity.