  Let $\mathscr{R}$ be a type $II_1$ von Neumann algebra. We show that every
unitary in $\mathscr{R}$ may be decomposed as the product of six symmetries
(that is, self-adjoint unitaries) in $\mathscr{R}$, and every unitary in
$\mathscr{R}$ with finite spectrum may be decomposed as the product of four
symmetries in $\mathscr{R}$. Consequently, the set of products of four
symmetries in $\mathscr{R}$ is norm-dense in the unitary group of
$\mathscr{R}$. Furthermore, we show that the set of products of three
symmetries in a von Neumann algebra $\mathscr{M}$ is not norm-dense in the
unitary group of $\mathscr{M}$. This strengthens a result of Halmos-Kakutani
which asserts that the set of products of three symmetries in
$\mathcal{B}(\mathscr{H})$, the ring of bounded operators on a Hilbert space
$\mathscr{H}$, is not the full unitary group of $\mathcal{B}(\mathscr{H})$.
