  The chirally rotated Schr\"odinger functional ($\chi$SF) renders the
mechanism of automatic $O(a)$ improvement compatible with Schr\"odinger
functional (SF) renormalization schemes. Here we define a family of
renormalization schemes based on the $\chi$SF for a complete basis of $\Delta F
= 2$ parity-odd four-fermion operators. We compute the corresponding
scale-dependent renormalization constants to one-loop order in perturbation
theory and obtain their NLO anomalous dimensions by matching to the
$\overline{\textrm{MS}}$ scheme. Due to automatic $O(a)$ improvement, once the
$\chi$SF is renormalized and improved at the boundaries, the step scaling
functions (SSF) of these operators approach their continuum limit with
$O(a^{2})$ corrections without the need of operator improvement.
