Switchbacks—rapid, large deflections of the solar wind's magnetic field—have generated interest as possible signatures of the mechanisms that heat the corona and accelerate the solar wind. In this context, an important task for theories of switchback formation and evolution is to understand their observable distinguishing features, allowing them to be carefully assessed using spacecraft data. Here, we work toward this goal by studying how Alfvénic switchbacks evolve in the expanding solar wind beyond the Alfvén radius, when the background magnetic field also rotates due to the Parker spiral. Using simple analytic arguments based on the physics of one-dimensional spherically polarized (constant-field-magnitude) Alfvén waves, we find that, by controlling the wave's obliquity, a Parker spiral strongly impacts switchback properties. Surprisingly, parallel magnetic-field deflections (switchbacks) can grow faster in a Parker spiral than in a radial background field, even though normalized wave amplitudes grow more slowly. In addition, switchbacks become strongly asymmetric: large switchbacks preferentially involve magnetic-field rotations in the plane of the Parker spiral (tangential deflections) rather than perpendicular (normal) rotations, and such deflections are strongly “tangentially skewed,” meaning switchbacks always involve field rotations in the same direction (toward the positive-radial direction for an outward mean field). In a companion paper [Johnston et al., Phys. Plasmas 29, 072902 1346 (2022)], we show that these properties also occur in turbulent 3D fields with switchbacks, given various caveats. Given that these nontrivial asymmetries and correlations develop purely as a consequence of switchback propagation in the solar wind, our results show that in situ observed asymmetrical switchback features cannot be used straightforwardly to infer properties of sources in the low corona.