We present optical spectropolarimetry Of three post-main-sequence bipolar nebulae: GL 618, MI-92, and M2-56. These objects represent the transitional stage between asymptotic giant branch stars and planetary nebulae. The optical spectra of their bipolar lobes are composed of a scattered stellar continuum plus emission lines. We use the technique of spectropolarimetry to separate line emission originating within the lobes (the '' unscattered spectrum '') from line emission reflected from the central regions (the '' scattered spectrum ''). The emission lines in GL 618 and Mq-92 are partially polrized and include both scattered and unscattered components; for M2-56, the emission lines are unpolarized. The unscattered spectra resemble the spectra of Herbig-Haro objects and are consistent with excitation by shocks with velocities in the range 40-100 km s-1. We use the derived unscattered line intensities, together with shock models in the current literature, to estimate ps densities, temperatures, N/O abundances, and preshock densities. GL 618 and possibly M2-56 apparently have elevated N/O ratios, a signature of nucleosynthetic processing followed by mixing, whereas the N/ratio in M1-92 appears consistent with the solar value. The derived preshock densities range from about 500-800 cm-3 for M1-92 to 2000-7000 cm-3 for GL 618. Our observations demonstrate that shocks dominate the observed line spectra of these bipolar nebulae; shocks may also play a major role in their morphology and kinematic evolution. We discuss our results in the context of various proposed models for the formation mechanism and evolutionary status of bipolar nebulae.