Uncountably many $\pm1$ sign patterns give unitary diagonal operators

For any sequence $(\alpha_n)$ of real numbers, define $T(\alpha_n):\ell^2\to\ell^2$ by $T(\alpha_n)(e_k)=\alpha_ke_k$. The number of unitary operators of the form $T(\alpha_n)$ on $\ell^2$ is: