  We present a framework for the construction of linearizations for scalar and
matrix polynomials based on dual bases which, in the case of orthogonal
polynomials, can be described by the associated recurrence relations. The
framework provides an extension of the classical linearization theory for
polynomials expressed in non-monomial bases and allows to represent polynomials
expressed in product families, that is as a linear combination of elements of
the form $\phi_i(\lambda) \psi_j(\lambda)$, where $\{ \phi_i(\lambda) \}$ and
$\{ \psi_j(\lambda) \}$ can either be polynomial bases or polynomial families
which satisfy some mild assumptions. We show that this general construction can
be used for many different purposes. Among them, we show how to linearize sums
of polynomials and rational functions expressed in different bases. As an
example, this allows to look for intersections of functions interpolated on
different nodes without converting them to the same basis. We then provide some
constructions for structured linearizations for $\star$-even and
$\star$-palindromic matrix polynomials. The extensions of these constructions
to $\star$-odd and $\star$-antipalindromic of odd degree is discussed and
follows immediately from the previous results.
