Summary: In this note we show that the set of operators, \(S=\{I, T, P, T\circ P\}\) that consists of the identity \(I\), the usual transpose \(T\), the per-transpose \(P\) and their product \(T\circ P\), forms a Klein Four-Group with the composition. With the introduced framework, we study in detail the properties of bisymmetric, centrosymmetric matrices and other algebraic structures, and we provide new definitions and results concerning these structures. In particular, we show that the per-tansposition allows to define a degenerate inner product of vectors, a cross product and a dyadic product of vectors with some interesting properties. In the last part of the work, we provide another realization of the Klein Group involving the tensorial product of some \(2 \times 2\) matrices.