This paper is devoted to a study of symmetric paraunitary matrix extensions. The problem for a given compactly supported orthonormal scaling vector with some symmetric property, to construct a corresponding multiwavelet which also has the symmetric property, is equivalent to the symmetric paraunitary extension of a given matrix. In this paper we study symmetric paraunitary extensions of two types of matrices which correspond to two different cases for the symmetry of the scaling vector: the components of the scaling vector have or don't have the same symmetric center. In this paper we also discuss parametrizations of symmetric orthogonal multifilter banks.