  Multiple orthogonality is considered in the realm of a Gauss--Borel
factorization problem for a semi-infinite moment matrix. Perfect combinations
of weights and a finite Borel measure are constructed in terms of M-Nikishin
systems. These perfect combinations ensure that the problem of mixed multiple
orthogonality has a unique solution, that can be obtained from the solution of
a Gauss--Borel factorization problem for a semi-infinite matrix, which plays
the role of a moment matrix. This leads to sequences of multiple orthogonal
polynomials, their duals and second kind functions. It also gives the
corresponding linear forms that are bi-orthogonal to the dual linear forms.
Expressions for these objects in terms of determinants from the moment matrix
are given, recursion relations are found, which imply a multi-diagonal Jacobi
type matrix with snake shape, and results like the ABC theorem or the
Christoffel--Darboux formula are re-derived in this context (using the
factorization problem and the generalized Hankel symmetry of the moment
matrix). The connection between this description of multiple orthogonality and
the multi-component 2D Toda hierarchy, which can be also understood and studied
through a Gauss--Borel factorization problem, is discussed. Deformations of the
weights, natural for M-Nikishin systems, are considered and the correspondence
with solutions to the integrable hierarchy, represented as a collection of Lax
equations, is explored. Corresponding Lax and Zakharov--Shabat matrices as well
as wave functions and their adjoints are determined. The construction of
discrete flows is discussed in terms of Miwa transformations which involve
Darboux transformations for the multiple orthogonality conditions. The bilinear
equations are derived and the $\tau$-function representation of the multiple
orthogonality is given.
