Call $A$ this matrix. If you assume in addition that $A$ is diagonalisable, then $A$ is a product of two symmetric matrices $S_+S$ where $S_+$ is positive definite and the signs of the eigenvalues of $S$ are the signs of the eigenvalues of $A$.

Of course, you don't expect that $A$ be unitarily (= orthogonaly here) to a symmetric matrix, because it would be itself symmetric.

As said by many, if $A$ is not diagonalisable, it cannot be similar to a symmetric matrix.

Finally, Theorem 4.1.7 in R. Horn & C. Johnson says that every real matrix is a product of two hermitian matrices, but I don't know whether you may take real factors.

Call $A$ this matrix. If you assume in addition that $A$ is diagonalisable, then $A$ is a product of two symmetric matrices $S_+S$ where $S_+$ is positive definite and the signs of the eigenvalues of $S$ are the signs of the eigenvalues of $A$.

Of course, you don't expect that $A$ be unitarily (= orthogonaly here) similar to a symmetric matrix, because it would be itself symmetric.

As said by many, if $A$ is not diagonalisable, it cannot be similar to a symmetric matrix.

Finally, Theorem 4.1.7 in R. Horn & C. Johnson says that every real matrix is a product of two hermitian matrices, but I don't know whether you may take real factors.