Consider $\mathbb C^n$ as the pseudounitary space with the inner product defined by the matrix $$ \mathcal P_n=\left(\begin{array}{cccc} &&&1\\ &&1&\\ &\cdots&&\\ 1&&& \end{array}\right). $$ In this space, centrounitary matrices play the role of unitary operators. The main result of this paper describes a certain factorization of an arbitrary centrounitary matrix of even order into a product of simpler centrounitary matrices. This result is an implication of a similar fact concerning factorizations of pseudounitary matrices of the type $(n,n)$.