  The chirally rotated Schr\"odinger functional ($\chi$SF) with massless
Wilson-type fermions provides an alternative lattice regularization of the
Schr\"odinger functional (SF), with different lattice symmetries and a common
continuum limit expected from universality. The explicit breaking of flavour
and parity symmetries needs to be repaired by tuning the bare fermion mass and
the coefficient of a dimension 3 boundary counterterm. Once this is achieved
one expects the mechanism of automatic O($a$) improvement to be operational in
the $\chi$SF, in contrast to the standard formulation of the SF. This is
expected to significantly improve the attainable precision for step-scaling
functions of some composite operators. Furthermore, the $\chi$SF offers new
strategies to determine finite renormalization constants which are
traditionally obtained from chiral Ward identities. In this paper we consider a
complete set of fermion bilinear operators, define corresponding correlation
functions and explain the relation to their standard SF counterparts. We
discuss renormalization and O($a$) improvement and then use this set-up to
formulate the theoretical expectations which follow from universality.
Expanding the correlation functions to one-loop order of perturbation theory we
then perform a number of non-trivial checks. In the process we obtain the
action counterterm coefficients to one-loop order and reproduce some known
perturbative results for renormalization constants of fermion bilinears. By
confirming the theoretical expectations, this perturbative study lends further
support to the soundness of the $\chi$SF framework and prepares the ground for
non-perturbative applications.

High Energy Physics - Lattice