A vertex operator superalgebra $V$ is called regular if every weak (no grading required) $V$-module is a direct sum of ordinary (with grading) irreducible modules. Let $g$ be a finite-order automorphism of $V$. One defines $g$-twisted modules by imposing the $g$@-twisted Jacobi identity. If all $g$-twisted weak modules are direct sums of irreducible $g$-twisted submodules, then $V$ is called $g$-regular. \par This paper studies the vertex operator superalgebra $L_{c_m}$ associated to the unitary vacuum module for the $N = 2$ superconformal algebra with the central charge ${c_m = \frac{3m}{m+2}}$, $m \in \Bbb{N}$. D. Adamović [J. Algebra {\bf 239} (2001), no.~2, 549--572; MR1832905] proved that $L_{c_m}$ is regular. One crucial step therein was to identify the tensor product of $L_{c_m}$ and a lattice vertex algebra with a simple current extension. Adapting these arguments to the twisted situation, the author shows that $L_{c_m}$ is $\sigma$-regular, where $\sigma(x) = \pm x$ depending on whether $x$ is even or odd.