Natural walking can provide a compelling experience in immersive virtual environments, but it remains an implementation challenge due to the physical space constraints imposed on the size of the virtual world. The use of redirection techniques is a promising approach that relaxes the space requirements of natural walking by manipulating the user's route in the virtual environment, causing the real world path to remain within the boundaries of the physical workspace. In this paper, we present and apply a novel taxonomy that separates redirection techniques according to their geometric flexibility versus the likelihood that they will be noticed by users. Additionally, we conducted a user study of three reorientation techniques, which confirmed that participants were less likely to experience a break in presence when reoriented using the techniques classified as subtle in our taxonomy. Our results also suggest that reorientation with change blindness illusions may give the impression of exploring a more expansive environment than continuous rotation techniques, but at the cost of negatively impacting spatial knowledge acquisition.