  Using the decomposition of semimagic squares into the associated and balanced
symmetry types as a motivation, we introduce an equivalent representation in
terms of block-structured matrices. This block representation provides a way of
constructing such matrices with further symmetries and of studying their
algebraic behaviour, significantly advancing and contributing to the
understanding of these symmetry properties. In addition to studying classical
attributes, such as dihedral equivalence and the spectral properties of these
matrices, we show that the inherent structure of the block representation
facilitates the definition of low-rank semimagic square matrices. This is
achieved by means of tensor product blocks. Furthermore, we study the rank and
eigenvector decomposition of these matrices, enabling the construction of a
corresponding two-sided eigenvector matrix in rational terms of their entries.
The paper concludes with the derivation of a correspondence between the tensor
product block representations and quadratic form expressions of Gaussian type.
