Supernova explosions are inherently asymmetric and can accelerate new-born neutron stars (NSs) to hundreds of km s$^{-1}$. Two prevailing theories to explain NS kicks are ejecta asymmetries (e.g., conservation of momentum between NS and ejecta) and anisotropic neutrino emission. Observations of supernova remnants (SNRs) can give us insights into the mechanism that generates these NS kicks. In this paper, we investigate the relationship between NS kick velocities and the X-ray morphologies of 18 SNRs observed with the Chandra X-ray Observatory and the Röntgen Satellite (ROSAT). We measure SNR asymmetries using the power-ratio method (a multipole expansion technique), focusing on the dipole, quadrupole, and octupole power ratios. Our results show no correlation between the magnitude of the power ratios and NS kick velocities, but we find that for Cas A and G292.0+1.8, whose emission traces the ejecta distribution, their NSs are preferentially moving opposite to the bulk of the X-ray emission. In addition, we find a similar result for PKS 1209-51, CTB 109, and Puppis A; however, their emission is dominated by circumstellar/interstellar material, so their asymmetries may not reflect their ejecta distributions. Our results are consistent with the theory that NS kicks are a consequence of ejecta asymmetries as opposed to anisotropic neutrino emission. In the future, additional observations to measure NS proper motions within ejecta-dominated SNRs are necessary to robustly constrain the NS kick mechanism.