We present a detailed imaging and spectroscopic study of the conjugate hard X-ray (HXR) footpoints (FPs) observed with the Ramaty High Energy Solar Spectroscopic Imager (RHESSI) in the 2003 October 29 X10 flare. The double FPs first move toward and then away from each other, mainly parallel and perpendicular to the magnetic neutral line, respectively. The transition of these two phases of FP unshearing motions coincides with the direction reversal of the motion of the loop-top (LT) source, and with the minima of the estimated loop length and LT height. We find temporal correlations between the HXR flux, spectral index, and magnetic field strength of each FP. The HXR flux exponentially correlates with the magnetic field strength, which also anticorrelates with the spectral index before the second HXR peak's maximum, suggesting that particle acceleration sensitively depends on the magnetic field strength and/or reconnection rate. Asymmetries are observed between the FPs: on average, the eastern FP is 2.2 times brighter in HXR flux and 1.8 times weaker in magnetic field strength, and moves 2.8 times faster away from the neutral line than the western FP; the estimated coronal column density to the eastern FP from the LT source is 1.7 times smaller. The two FPs have marginally different spectral indices. The eastern-to-western FP HXR flux ratio and magnetic field strength ratio are anticorrelated only before the second HXR peak's maximum. Neither magnetic mirroring nor column density alone can explain the totality of these observations, but their combination, together with other transport effects, might provide a full explanation. We have also developed novel techniques to remove particle contamination from HXR counts and to estimate effects of pulse pileup in imaging spectroscopy, which can be applied to other RHESSI flares in similar circumstances.