Weather radar moments and polarimetric variables provide useful information about the characteristics and motion of hydrometeors. However, the bulk information may be masked when the meteorological signal of interest is contaminated by clutter. The dual-polarimetric spectral densities (DPSD) may unveil additional information about the polarimetric characteristics of groups of scatterers moving at different Doppler velocities in a given radar resolution volume. Previous DPSD estimation methods required averaging a large number of spectra (obtained from different range gates, radials, or scans), or averaging in frequency to get accurate estimates; though by doing so, the resolution is degraded, and important features of the meteorological phenomenon may be masked, potentially affecting the ability to perform a good spectral analysis. In an attempt to overcome these limitations, the Bootstrap DPSD estimator is developed, which allows the estimation of DPSDs from a single dwell, with minimal resolution loss. Briefly, the estimator pre-processes the weather radar I/Q time-series signals and generates I/Q pseudo-realizations through bootstrap resampling, which are then used to compute PSD estimates that are averaged to obtain the DPSD estimate. Then, a post-processing stage applies a bias correction to the estimates. The Bootstrap DPSD estimator's performance is compared to that of conventional methods for single-dwell as well as for multiple-dwell estimates. Additionally, the performance and limitations of the Bootstrap and conventional DPSD estimators are assessed when identifying signals of different polarimetric signatures of scatterers moving at different radial velocities in the radar volume. The advantages of the Bootstrap DPSD estimator as a tool for polarimetric spectral analysis is demonstrated with a few examples of polarimetric spectral signatures in data from tornado cases, and from a physically-based simulator. It is expected that, with the Bootstrap DPSD and polarimetric spectral analysis, it will be possible to better understand tornado dynamics and their connection to weather radar measurements, as well as to elucidate important scientific questions that motivated this work.