- \psi^{1-2}(\sigma), \\
\psi^{3-4}(\sigma + 2 \pi) & = & - \psi^{3-4}(\sigma), \\
\lambda^{1-6}(\sigma + 2 \pi) & = & + \lambda^{1-6}(\sigma), \\
\lambda^{7-14}(\sigma + 2 \pi) & = & + \exp\left( 2 \pi i
\frac{2}{4} \right) \lambda^{7-14}(\sigma) \: = \:
- \lambda^{7-14}(\sigma), \\
\lambda^{15-16}(\sigma + 2 \pi) & = & + \lambda^{15-16}(\sigma).\end{aligned}$$ It is straightforward to compute that $E_{\rm left} = 0$, $E_{\rm right} = -1/2$. There is a multiplicity of left vacua, as $\lambda^{1-6}$ and $\lambda^{15-16}$ are periodic. In particular, $|\pm \mp \rangle_{15-16}$ are invariant under ${\mathbb Z}_4$, whereas $|\pm \pm \rangle_{15-16}$ get a sign flip. Therefore, the ${\mathbb Z}_4$-invariant massless states are of the form

  -------------------------------------------------------------------------------------------------------------
                            State                                                  Count
  --------------------------------------------------------- ---------------------------------------------------
               $| \pm \cdots \pm \rangle_{1-6}                               spacetime vector
                  | \pm \mp \rangle_{15-16}                 
                           \otimes                          
    \left( \psi^{1-2}_{-1/2}, \overline{\psi}^{1-2}_{-1/2}  
                          \right)$                          

                                                              in $({\bf 32},{\bf 2})$ of $so(12)\times so(4)$

   $| \pm \cdots \pm\rangle_{1-6} |\pm \pm \rangle_{15-16}                   4 sets of scalars
                           \otimes                          
   \left( \psi^{3-4}_{-1/2}, \overline{\psi}^{3-4}_{-1/2}   
                          \right)$                          

                                                             in $({\bf 32}',{\bf 2}')$ of $so(12)\times so(4)$
  -------------------------------------------------------------------------------------------------------------

We can rearrange the spacetime vecto