Redirected walking (RDW) facilitates user navigation within expansive virtual spaces despite the constraints of limited physical spaces. It employs discrepancies between human visual-proprioceptive sensations, known as gains, to enable the remapping of virtual and physical environments. In this paper, we explore how to apply rotation gain while the user is walking. We propose to apply a rotation gain to let the user rotate by a different angle when reciprocating from a previous head rotation, to achieve the aim of steering the user to a desired direction. To apply the gains imperceptibly based on such a Bidirectional Rotation gain Difference (BiRD), we conduct both measurement and verification experiments on the detection thresholds of the rotation gain for reciprocating head rotations during walking. Unlike previous rotation gains which are measured when users are turning around in place (standing or sitting), BiRD is measured during users' walking. Our study offers a critical assessment of the acceptable range of rotational mapping differences for different rotational orientations across the user's walking experience, contributing to an effective tool for redirecting users in virtual environments.