  Many applications give rise to structured matrix polynomials. The problem of
constructing structure-preserving strong linearizations of structured matrix
polynomials is revisited in this work and in the forthcoming ones
\cite{PartII,PartIII}. With the purpose of providing a much simpler framework
for structure-preserving linearizations for symmetric and skew-symmetric matrix
polynomial than the one based on Fiedler pencils with repetition, we introduce
in this work the families of (modified) symmetric and skew-symmetric block
Kronecker pencils. These families provide a large arena of structure-preserving
strong linearizations of symmetric and skew-symmetric matrix polynomials. When
the matrix polynomial has degree odd, these linearizations are strong
regardless of whether the matrix polynomial is regular or singular, and many of
them give rise to structure-preserving companion forms. When some generic
nonsingularity conditions are satisfied, they are also strong linearizations
for even-degree regular matrix polynomials. Many examples of
structure-preserving linearizations obtained from Fiedler pencils with
repetitions found in the literature are shown to belong (modulo permutations)
to these families of linearizations. In particular, this is shown to be true
for the well-known block-tridiagonal symmetric and skew-symmetric companion
forms. Since the families of symmetric and skew-symmetric block Kronecker
pencils belong to the recently introduced set of minimal bases pencils
\cite{Fiedler-like}, they inherit all its desirable properties for numerical
applications. In particular, it is shown that eigenvectors, minimal indices,
and minimal bases of matrix polynomials are easily recovered from those of any
of the linearizations constructed in this work.
